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Revision as of 06:57, 24 April 2012
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| Table of Infinite Integrals | |
|---|---|
| 1 General Rules | |
| $ \int a d x = a x $ | |
| $ \int a f ( x ) d x = a \int f ( x ) d x $ | |
| $ \int ( u \pm v \pm w \pm \cdot \cdot \cdot ) d x = \int u d x \pm \int v d x \pm \int w d x \pm \cdot \cdot \cdot $ | |
| $ \int u d v = u v - \int v d u $ | |
| $ \int f ( a x ) d x = \frac{1}{a} \int f ( u ) d u $ | |
| $ \int F \{ f ( x ) \} d x = \int F ( u ) \frac{dx}{du} d u = \int \frac{F ( u )}{f^' ( x )} d u \qquad u = f ( x ) $ | |
| $ \int u^n d u = \frac{u^{n+1}}{n+1} \qquad n \neq -1 $ | |
| $ \int \frac{d u}{u} = \ln u \ ( if \ u > 0 ) \ \text{or} \ln {-u} \ ( \text{if} \ u < 0 ) = \ln \left | u \right | $ | |
| $ \int e^u d u = e^u $ | |
| $ \int a^u d u = \int e^{u \ln a} d u = \frac{e^{u \ln a}}{\ln a} = \frac{a^u}{\ln a} \qquad a > 0 \ \text{and} \ a \neq 1 $ | |
| $ \int \sin u\ d u = - \cos u $ | |
| $ \int \cos u\ d u = \sin u $ | |
| $ \int \tan u\ d u = - \ln {\cos u} $ | |
| $ \int \cot u\ d u = \ln {\sin u} $ | |
| $ \int \frac{d u}{\cos u} = \ln { \left ( \frac{1}{\cos u} + \tan u \right )} = \ln{\tan {\left ( \frac{u}{2}+\frac{\pi}{4}\right )}} $ | |
| $ \int \frac{d u}{\sin u} = \ln { \left ( \frac{1}{\sin u} - \cot u \right )} = \ln{\tan { \frac{u}{2}}} $ | |
| $ \int \frac{d u}{\cos ^2 u} = \tan u $ | |
| $ \int \frac{d u}{\sin ^2 u} = - \cot u $ | |
| $ \int \tan ^2 u \ d u = \tan u - u $ | |
| $ \int \cot ^2 u \ d u = - \cot u - u $ | |
| $ \int \sin ^2 u \ d u= \frac{u}{2} - \frac{\sin {2 u}}{4} = \frac{1}{2}\left( u - \sin u \cos u \right ) $ | |
| $ \int \frac {1}{\cos u} \tan u \ d u = \frac{1}{\cos u} $ | |
| $ \int \frac {1}{\sin u} \cot u \ d u = - \frac{1}{\sin u} $ | |
| $ \int \sinh u \ d u = \coth u $ | |
| $ \int \cosh u \ d u = \sinh u $ | |
| $ \int \tanh u \ d u = \ln \cosh u $ | |
| $ \int \coth u \ d u = \ln \sinh u $ | |
| $ \int \frac {1}{\operatorname{ch}\ u} \ d u = \arcsin{\left ( \operatorname{th}\,u \right )} \qquad or \ 2 arc \ th \ e^u $ | |
| $ \int \frac {1}{\operatorname{sh}\ u} \ d u = \ln \operatorname{th}\,\frac{2}{2} \qquad or \ - \operatorname{Arg coth} \ e^u $ | |
| $ \int \frac {1}{\operatorname{ch^2}\ u} \ d u = \operatorname{th}\,u $ | |
| $ \int \frac {1}{\operatorname{sh^2}\ u} \ d u = - \operatorname{coth}\,u $ | |
| $ \int \operatorname{th^2}\ u \ d u = u - \operatorname{th}\,u $ | |
| $ \int \operatorname{coth^2}\ u \ d u = u - \operatorname{coth}\,u $ | |
| $ \int \operatorname{sh^2}\ u \ d u = \frac {\operatorname{sh}\,{2 u}}{4} - \frac{u}{2}=\frac{1}{2}\left ( \operatorname{sh}\,u \ \operatorname{ch}\,u - u \right ) $ | |
| $ \int \operatorname{ch^2}\ u \ d u = \frac {\operatorname{sh}\,{2 u}}{4} + \frac{u}{2}=\frac{1}{2}\left ( \operatorname{sh}\,u \ \operatorname{ch}\,u + u \right ) $ | |
| $ \int \frac{\operatorname th \ u}{\operatorname ch \ u} \ d u = - \frac {1}{\operatorname ch \, u } $ | |
| $ \int \frac{\operatorname coth \ u}{\operatorname sh \ u} \ d u = - \frac {1}{\operatorname sh \, u } $ | |
| $ \int \frac{d u}{u^2 + a^2} = \frac {1}{a}\arctan \frac{u}{a} $ | |
| $ \int \frac{d u}{u^2 - a^2} = \frac {1}{2 a}\ln \left ( \frac{u-a}{u+a} \right ) = -\frac{1}{a} \operatorname{argcoth} \ \frac{u}{a} \qquad u^2 > a^2 $ | |
| $ \int \frac{d u}{a^2 - u^2} = \frac {1}{2 a}\ln \left ( \frac{a+u}{a-u} \right ) = \frac{1}{a} \operatorname{argth}\ \frac{u}{a} \qquad u^2 < a^2 $ | |
| $ \int \frac{d u}{\sqrt{a^2 - u^2}} = \arcsin \frac{u}{a} $ | |
| $ \int \frac{d u}{\sqrt{u^2 + a^2}} = \ln { \left ( u + \sqrt {u^2+a^2} \right ) } \qquad or \ \operatorname{argth} \ \frac{u}{a} $ | |
| $ \int \frac{d u}{\sqrt{u^2 - a^2}} = \ln { \left ( u + \sqrt {u^2-a^2} \right ) } $ | |
| $ \int \frac{d u}{u \sqrt{u^2 - a^2}} = \frac {1}{a} \arccos \left | \frac{a}{u} \right | $ | |
| $ \int \frac{d u}{u \sqrt{u^2 + a^2}} = - \frac {1}{a} \ln \left ( \frac{a + \sqrt{u^2 + a^2}}{u} \right ) $ | |
| $ \int \frac{d u}{u \sqrt{a^2 - u^2}} = - \frac {1}{a} \ln \left ( \frac{a + \sqrt{a^2 - u^2}}{u} \right ) $ | |
| $ \int f^{(n)} \ g d x =f^{(n-1)} \ g - f^{(n-2)} \ g' + f^{(n-3)} \ g'' - \cdot \cdot \cdot \ (-1)^n \int fg^{(n)} d x $ | |
| 2 Important Transformations | |
| $ \int F( a x + b) d x =\frac{1}{a} \int F( u) d u \qquad u = a x + b $ | |
| $ \int F( \sqrt {a x + b} ) d x =\frac{2}{a} \int u F( u) d u \qquad u = \sqrt {a x + b} $ | |
| $ \int F( \sqrt [n] {a x + b} ) d x = \frac{n}{a} \int u^{n-1} F( u) d u \qquad u = \sqrt [n] {a x + b} $ | |
| $ \int F( \sqrt {a^2 - x^2} ) d x =a \ \int F( a \cos u) \ \cos u \ d u \qquad x = a \sin u $ | |
| $ \int F( \sqrt {x^2 + a^2} ) d x =a \ \int F \left ( \frac {a}{\cos u} \right ) \frac {1}{\cos ^2 u} \ d u \qquad x = a \tan u $ | |
| $ \int F( \sqrt {x^2 - a^2} ) d x =a \ \int F \left ( a \tan u \right ) \frac {\tan u}{\cos u} \ d u \qquad x = \frac {a}{\cos u} $ | |
| $ \int F( e ^{a x}) d x = \frac {1}{a} \int \frac {F(u)}{u} \ d u \qquad u = e^{a x} $ | |
| $ \int F( \ln x ) d x = \int F(u)\ e^u \ d u \qquad u = \ln x $ | |
| $ \int F\left ( \arcsin \frac{x}{a} \right) d x = a \int F(u)\ \cos u \ d u \qquad u = \arcsin \frac {x}{a} $ | |
| $ \int F\left ( \sin x ,\cos x \right) d x = 2 \int F \left( \frac {2 u}{1 + u^2}, \frac {1 - u^2}{1+u^2} \right)\ \frac {d u}{1+ u^2} \qquad u = \tan \frac {x}{2} $ | |
| 3 Particular Integral, component ax +b | |
| $ \int \frac {d x}{ ax + b} = \frac {1}{a} \ln (ax +b) $ | |
| $ \int \frac {x d x}{ ax + b} = \frac {x}{a} - \frac{b}{a^2} \ln (ax +b) $ | |
| $ \int \frac {x^2 d x}{ ax + b} = \frac {(ax+b)^2}{2a^3} - \frac {2b(ax+b) }{a^3} + \frac{b^2}{a^3} \ln (ax +b) $ | |
| $ \int \frac {x^3 d x}{ ax + b} = \frac {(ax+b)^3}{3a^4} - \frac {3b(ax+b)^2 }{2a^4} + \frac{3b^2(ax+b)}{a^4} - frac{b^3}{a^3}\ln (ax +b) $ | |
| $ \int \frac {d x}{ x(ax + b)} = \frac {1}{b} \ln \left ( \frac {x}{ax +b} \right) $ | |
| $ \int \frac {d x}{ x^2(ax + b)} = - \frac {1}{b x} + \frac {a}{b^2} \ln \left ( \frac {ax +b}{x} \right) $ | |
| $ \int \frac {d x}{ x^3(ax + b)} = \frac {2 a x - b}{2 b^2 x^2} + \frac {a^2}{b^3} \ln \left ( \frac {x}{ax+b} \right) $ | |
| $ \int \frac {d x}{(ax + b)^2} = \frac {-1}{a(ax+b)} $ | |
| $ \int \frac {x d x}{(ax + b)^2} = \frac {b}{a^2(ax+b)} + \frac {1}{a^2} \ln (ax+b) $ | |
| $ \int \frac {x^2 d x}{(ax + b)^2} = \frac {ax+b}{a^3} - \frac{b^2}{a^3(ax+b)} - \frac {2b}{a^3} \ln (ax+b) $ | |
| $ \int \frac {x^3 d x}{(ax + b)^2} = \frac {(ax+b)^2}{2a^4} - \frac {3b(ax+b)}{a^4} +\frac{b^3}{a^4(ax+b)} + \frac {3b^2}{a^4} \ln (ax+b) $ | |
| $ \int \frac {d x}{x(ax + b)^2} = \frac {1}{b(ax+b)} + \frac {1}{b^2} \ln \left ( \frac{x}{ax+b} \right ) $ | |
| $ \int \frac {d x}{x^2(ax + b)^2} = \frac {-a}{b^2(ax+b)} - \frac {1}{b^2x} + \frac {2a}{b^3} \ln \left ( \frac {ax+b}{x} \right ) $ | |
| $ \int \frac {d x}{x^3(ax + b)^2} = - \frac {(ax+b)^2}{2b^4x^2} + \frac {3 a(ax+b)}{b^4x} - \frac {a^3 x}{b^4(ax+b)} - \frac{3a^2}{b^4} \ln \left ( \frac {ax+b}{x} \right ) $ | |
| $ \int \frac {d x}{(ax + b)^3} = \frac {-1}{2(ax+b)^2} $ | |
| $ \int \frac {x d x}{(ax + b)^3} = \frac {-1}{a^2(ax+b)} + \frac {b}{2a^2(ax+b)^2} $ | |
| $ \int \frac {x^2 d x}{(ax + b)^3} = \frac {2b}{a^3(ax+b)} - \frac {b^2}{2a^3(ax+b)^2} + \frac {1}{a^3} \ln (ax+b) $ | |
| $ \int \frac {x^3 d x}{(ax + b)^3} = \frac {x}{a^3} - \frac {3b^2}{a^4(ax+b)} + \frac {b^3}{2a^4(ax+b)^2} - \frac {3b}{a^4} \ln (ax+b) $ | |
| $ \int \frac {d x}{x(ax + b)^3} = \frac {a^2x^2}{2b^3(ax+b)^2} - \frac {2ax}{b^3(ax+b)} - \frac {1}{b^3} \ln \left( \frac{ax+b}{x} \right) $ | |
| $ \int \frac {d x}{x^2(ax + b)^3} = \frac {-a}{2b^2(ax+b)^2} - \frac {2a}{b^3(ax+b)} - \frac {1}{b^3x} + \frac {3a}{b^4} \ln \left( \frac{ax+b}{x} \right) $ | |
| $ \int \frac {d x}{x^3(ax + b)^3} = \frac {a^4x^2}{2b^5(ax+b)^2} - \frac {4a^3x}{b^5(ax+b)} - \frac {(ax+b)^2}{2b^5x2} - \frac {6a^2}{b^5} \ln \left( \frac{ax+b}{x} \right) $ | |
| $ \int (a x +b)^n d x = \frac {(ax+b)^{n+1} }{(n+1)a}. \qquad n =-1 $ | |
| $ \int x (a x +b)^n d x = \frac {(ax+b)^{n+2} }{(n+2)a^2} - \frac {b(ax+b)^{n+1}}{(n+1)a^2}, \qquad n \neq -1,-2 $ | |
| $ \int x^2 (a x + b)^n d x = \frac {(ax+b)^{n+3} }{(n+3)a^3} - \frac {2b(ax+b)^{n+2}}{(n+2)a^3} + \frac {b^2(ax+b)^{n+1}}{(n+1)a^3} \qquad n = -1,-2, -3 $ | |
| $ \int x^m (a x + b)^n d x = \begin{cases} \frac {x^{m+1}(ax+b)^n}{m + n + 1} + \frac {n b}{m + n+ 1} \int x^m (ax+b)^{n-1} d x \\ \frac {x^m(ax+b)^{n+1}}{(m + n + 1)a} - \frac {m b}{(m + n+ 1)a} \int x^{m-1} (ax+b)^{n} d x \\ \frac {- x^{m+1}(ax+b)^{n+1}}{(n + 1)b} + \frac {m+ n+ 2 }{(n+ 1)b} \int x^m (ax+b)^{n+1} d x \end{cases} $ | |
| $ \text{ 4 Particular Integral, component } \sqrt{ax +b} $ | |
| $ \int \frac {d x}{\sqrt{a x +b}} = \frac {2\sqrt{ax+b}}{a} $ | |
| $ \int \frac {x d x}{\sqrt{a x + b}} = \frac {2(ax-2b)}{3a^2}\sqrt{ax+b} $ | |
| $ \int \frac {x^2 d x}{\sqrt{a x + b}} = \frac {2(3a^2x^2-4abx + 8b^2)}{15a^3}\sqrt{ax+b} $ | |
| $ \int \frac {d x}{x \sqrt {ax+b}} = \begin{cases} \frac {1}{b} \ln \left ( \frac {\sqrt {ax+b} - \sqrt {b}}{\sqrt {ax+b} + \sqrt {b}} \right ) \\ \frac {2}{\sqrt {-b}} \arctan \sqrt { \frac {ax+b}{- b}} \\ \end{cases} $ | |
| $ \int \frac { d x}{x ^2 \sqrt{a x + b}} = - \frac {\sqrt{ax+b}}{b x} - \frac {a}{2 b} \int \frac {d x}{x \sqrt {ax + b}} $ | |
| $ \int \sqrt{a x + b} \ d x = \frac {2 \sqrt{(ax+b)3}}{3 a} $ | |
| $ \int x \sqrt{a x + b} \ d x = \frac {2(3ax-2b)}{15a^2}\sqrt{(ax+b)^3} $ | |
| $ \int x^2 \sqrt{a x + b} \ d x = \frac {2(15a^2x^2-12abx + 8b^2)}{105a^3}\sqrt{(ax+b)^3} $ | |
| $ \int \frac {\sqrt {ax+b}}{x} \ d x = 2 \sqrt {ax+b} + b \ \int \frac {d x}{x \sqrt {ax + b}} $ | |
| $ \int \frac {\sqrt {ax+b}}{x^2} \ d x = - \frac {\sqrt {ax+b}}{x} + \frac {a}{2} \int \frac {d x}{x \sqrt {ax + b}} $ | |
| $ \int \frac {x^m}{\sqrt{ ax+b}} d x = \frac {2x^m \sqrt {ax+b}}{(2m+1)a} - \frac {2mb}{(2m+1)a} \int \frac {x^{m-1}}{\sqrt {ax+b}} d x $ | |
| $ \int \frac {d x}{x^m \sqrt{ax+b}} =- \frac {\sqrt{ax+b}}{(m-1)bx^{m-1}} - \frac {(2m-3)a}{(2m-2)b} \int \frac {d x}{x^{m-1} \sqrt{ax+b}} $ | |
| $ \int x^m \sqrt {ax+b} \ d x = \frac{2x^m}{(2m+3)a}(a+b)^{\frac{3}{2}} -\frac {2mb}{(2m+3)a} \int x^{m-1} \sqrt{ax+b} \ d x $ | |
| $ \int \frac {\sqrt {ax+b}}{x^m} d x = - \frac {\sqrt {ax+b}}{(m-1)x^{m-1}} + \frac {a}{2(m-1)} \int \frac {d x}{x^{m-1} \sqrt {ax+b}} $ | |
| $ \int \frac {\sqrt {ax+b}}{x^m} d x = \frac {-(ax+b)^{3/2}}{(m-1)bx^{m-1}} - \frac {(2m-5)a}{(2m-2)b} \int \frac {\sqrt {ax+b}}{x^{m-1}} d x $ | |
| $ \int (ax+b)^{m/2} d x = \frac {2(ax+b)^{(m+2)/2}}{a(m+2)} $ | |
| $ \int x(ax+b)^{m/2} d x = \frac {2(ax+b)^{(m+4)/2}}{a^2(m+4)} - \frac {2b(ax+b)^{(m+2)/2}}{a^2(m+2)} $ | |
| $ \int x^2(ax+b)^{m/2} d x = \frac {2(ax+b)^{(m+6)/2}}{a^3(m+6)} - \frac {4b(ax+b)^{(m+4)/2}}{a^3(m+4)}+ \frac {2b^2(ax+b)^{(m+2)/2}}{a^3(m+2)} $ | |
| $ \int \frac {(ax+b)^{m/2}}{x} d x =\frac {2(ax+b)^{m/2}}{m} + b \ \int \frac {(ax+b)^{(m-2)/2}}{x} d x $ | |
| $ \int \frac {(ax+b)^{m/2}}{x^2} d x = - \frac {(ax+b)^{(m+2)/2}}{bx} + \frac {ma}{2b} \ \int \frac {(ax+b)^{m/2}}{x} d x $ | |
| $ \int \frac {d x}{x(ax+b)^{m/2}} d x = \frac {2}{(m-2)b(ax+b)^{(m-2)/2}} + \frac {1}{b} \ \int \frac {d x}{x(ax+b)^{(m-2)/2}} $ | |
| 5 Particular Integral, component ax + b and px + q | |
| $ \int \frac {d x}{(ax+b)(px+q)} = \frac {1}{bp-aq} \ln \left ( \frac {px+q}{ax+b} \right ) $ | |
| $ \int \frac {x d x}{(ax+b)(px+q)} = \frac {1}{bp-aq} \left \{ \frac{b}{a} \ln { (ax+b)} - \frac{q}{p} \ln{(px+q)} \right \} $ | |
| $ \int \frac { d x}{(ax+b)^2(px+q)} = \frac {1}{bp-aq} \left \{ \frac {1}{ax+b} + \frac {p}{bp- aq} \ln \left ( \frac {px+q}{ax +b} \right ) \right \} $ | |
| $ \int \frac {x d x}{(ax+b)^2(px+q)} = \frac {1}{bp-aq} \left \{ \frac {q}{bp- aq} \ln \left ( \frac {ax+b}{px +q} \right ) - \frac {b}{a(ax+b)} \right \} $ | |
| $ \int \frac {x^2 d x}{(ax+b)^2(px+q)} = \frac {b^2}{(bp -aq)a^2(ax+b)} + \frac {1}{(bp-aq)^2} \left \{ \frac {q^2}{p} \ln (px+q) + \frac {b(bp-2aq)}{a^2} \ln (ax+b) \right \} $ | |
| $ \int \frac { d x }{(ax+b)^m (px+q)^n } = \frac {-1}{(n-1)(bp-aq)} \left \{ \frac {1}{(ax+b)^{m-1}(pz+q)^{n-1} } + a(m + n -2) \ \int \frac {d x}{(ax+b)^m(px+q)^{n-1} } \right \} $ | |
| $ \int \frac {ax+b}{px+q} d x = \frac {ax}{p} + \frac {bp -aq}{p^2} \ln (px+q) $ | |
| $ \int \frac {(ax+b)^m}{(px+q)^n} d x = \begin{cases} \frac {-1}{(n-1)(bp-aq)} \left \{ \frac {(ax+b)^{m+1}}{(px+q)^{n-1}} + (n-m-2)a \int \frac {(ax+b)^m}{(px+q)^{n-1}} d x \right \} \\ \frac {-1}{(n-m-1)p} \left \{ \frac {(ax+b)^m}{(px+q)^{n-1}} + m(bp-aq) \int \frac{(ax+b)^{m-1}} {(px+q)^n} d x \right \} \\ \frac{-1}{(n-1)p} \left \{ \frac{(ax+b)^m}{(px+q)^{n-1}} - ma \int \frac{(ax+b)^{m-1}}{(px+q)^{n-1}} d x \right \} \end{cases} $ | |
| $ \text{ 6 Particular Integral, component } \sqrt{ax +b} \ and \ px+q $ | |
| $ \int \frac {px+q}{\sqrt {ax+b}} d x = \frac {2(apx+3aq-2bp)}{3a^2} \sqrt {ax+b} $ | |
| $ \int \frac {d x}{(px+q) \sqrt {ax+b}} = \begin{cases} \frac{1}{ \sqrt {bp-aq} \sqrt {p} } \ln \left ( \frac {\sqrt{p(ax+b)} -\sqrt {bp-aq} }{\sqrt{p(ax+b)} + \sqrt {bp-aq} } \right ) \\ \frac {2}{\sqrt {aq-bp} \sqrt {p}} \arctan \sqrt{ \frac {p(ax+b)}{aq-bp} } \end{cases} $ | |
| $ \int \frac {\sqrt {ax+b}}{px+q} \ d x = \begin{cases} \frac{2 \sqrt{ax+b}}{p} \ + \ \frac {\sqrt {bp-aq}}{p \sqrt{p}} \ln \left ( \frac {\sqrt {p(ax+b)} - \sqrt {bp-aq}}{\sqrt {p(ax+b)} + \sqrt {bp-aq}} \right ) \\ \frac {2 \sqrt {ax+b}}{p} \ - \ \frac {2 \sqrt{aq-bp}}{p \sqrt{p}} \arctan \sqrt { \frac {p(ax+b)}{aq-bp}} \\ \end{cases} $ | |
| $ \int (px+q)^n \sqrt {ax+b} \ d x = \frac{2(px+q)^{n+1} \sqrt {ax+b}}{(2n+3)p} \ + \ \frac {bp-aq}{(2n+3)p} \int \frac {(px+q)^n}{\sqrt {ax+b}} d x $ | |
| $ \int \frac {d x}{ (px+q)^n \sqrt {ax+b}} = \frac {\sqrt {ax+b}}{(n-1)(aq-bp)(px+q)^{n-1}} + \frac {(2n-3)a}{2(n-1)(aq-bp)} \int \frac {d x }{(px+q)^{n-1} \sqrt{ax+b}} $ | |
| $ \int \frac {(px+q)^n}{\sqrt {ax+b}} d x = \frac {2(px+q)^n \sqrt{ax+b}}{)2n+1)a} + \frac {2n(aq-bp)}{(2n+1)a} \int \frac {(px+q)^{n-1} d x}{\sqrt{ax+b}} $ | |
| $ \int \frac {\sqrt{ax+b}}{(px+q)^n} d x = \frac {- \sqrt {ax+b}}{(n-1)p(px+q)^n-1} + \frac{a}{2(n-1)p} \int \frac {d x}{(px+q)^{n-1} \sqrt {ax+b}} $ | |
| $ \text{7 Integeral Componant } \sqrt{ax+b} \text{ and } \sqrt{px+q} $ | |
| $ \int \frac {d x}{\sqrt {(ax+b)(px+q)}} = \begin{cases} \frac{2}{\sqrt{ap}} \ln { \left ( \sqrt {a(px+q)}+ \sqrt{p(ax+b)} \right )} \\ \frac {2}{\sqrt { -ap}} \arctan \ \sqrt { \frac {-p(ax+b)} {a(px+q)} } \\ \end{cases} $ | |
| $ \int \frac {x d x}{\sqrt {(ax+b)(px+q)}} = \frac {\sqrt{(ax+b)(px+q)}}{ap} - \frac{bp+aq}{2ap} \int \frac {d x}{\sqrt{(ax+b)(px+q)}} $ | |
| $ \int \sqrt {(ax+b)(px+q)} dx = \frac {2apx + bp +aq}{4ap} \sqrt {(ax+b)(px+q)} - \frac {(bp-aq)^2}{8ap} \int \frac{d x}{\sqrt{(ax+b)(px+q)}} $ | |
| $ \int \sqrt {\frac {px+q}{ax+b}} dx = \frac {\sqrt{(ax+b)(px+q)}}{a} + \frac {aq-bp}{2a} \int \frac {d x}{ \sqrt {(ax+b)(px+q)}} $ | |
| $ \int \frac {d x}{(px+q)\sqrt{(ax+b)(px+q)}} = \frac {2\sqrt{ax+b}}{(aq-bp)\sqrt{px+q}} $ | |
| $ \text {8 Integeral Componant } x^2 + a^2 $ | |
| $ \int \frac {d x}{ x^2 + a^2} \ = \ \frac {1}{a} \arctan \ \frac {x}{a} $ | |
| $ \int \frac {x \ d x}{x^2 + a^2} = \frac {1}{2} \ln \left ( x^2 + a^2 \right ) $ | |
| $ \int \frac {x^2 \ d x }{x^2 + a^2} = x \ - \ a \arctan \frac {x}{a} $ | |
| $ \int \frac {x^3 \ d x}{x^2 + a^2} = \frac{x^2}{2} - \frac{a^2}{2} \ln (x^2+a^2) $ | |
| $ \int \frac {d x}{ x(x^2 + a^2)} = \frac {1}{2a^2} \ln \left ( \frac {x^2}{x^2 + a^2} \right ) $ | |
| $ \int \frac {d x}{x^2(x^2 + a^2)} = - \frac {1}{a^2x} - \frac {1}{a^3} \arctan \frac {x}{a} $ | |
| $ \int \frac {d x}{x^3(x^2 + a^2)} = - \frac {1}{2a^2x^2} - \frac {1}{2a^4} \ln \left ( \frac {x^2}{x^2 + a^2} \right ) $ | |
| $ \int \frac {d x}{(x^2 +a^2)^2} =\frac {x}{2a^2(x^2+a^2)} + \frac {1}{2a^3} \arctan \frac {x}{a} $ | |
| $ \int \frac {x d x}{(x^2 +a^2)^2} = \frac {-1}{2(x^2 + a^2)} $ | |
| $ \int \frac {x^2 d x}{(x^2 +a^2)^2} = \frac {-x}{2(x^2 + a^2)} + \frac {1}{2a} \arctan \frac {x}{a} $ | |
| $ \int \frac {x^3 d x}{(x^2 +a^2)^2} = \frac {a^2}{2(x^2 + a^2)} + \frac{1}{2} \ln(x^2 + a^2) $ | |
| $ \int \frac {d x}{ x(x^2 + a^2)^2} = \frac {1}{2a^2(x^2+a^2)} + \frac{1}{2a^4} \ln \left ( \frac {x^2}{x^2 + a^2} \right ) $ | |
| $ \int \frac {d x}{x^2(x^2 + a^2)^2} = - \frac {1}{a^4x} - \frac {x}{2a^4(x^2 + a^2)} - \frac {3}{2a^5} \arctan \frac {x}{a} $ | |
| $ \int \frac {d x}{ x^3(x62 +a^2)^2} = - \frac {1}{2a^4x^2} - \frac {1}{2a^4(x^2+a^2)} - \frac {1}{a^6} \ln \left ( \frac {x^2}{x^2 + a^2} \right ) $ | |
| $ \int \frac {d x}{(x^2 + a^2)^n} = \frac {x}{2(n-1)a^2(x^2 + a^2)^{n-1}} + \frac {2n -3}{(2n - 2)a^2} \int \frac {d x}{(x^2 + a^2)^{n-1}} $ | |
| $ \int \frac {x dx}{(x^2+a^2)^n} = \frac {-1}{2(n-1)(x^2 + a^2)^{n-1}} $ | |
| $ \int \frac {d x}{x(x^2 +a^2)^n} = \frac {1}{2(n-1)a^2(x^2+a^2)^{n-1}} + \frac {1}{a^2} \int \frac {d x}{x(x^2 + a^2)^{n-1}} $ | |
| $ \int \frac {x^m d x}{(x^2 + a^2)^n} = \int \frac {x^{m-2} d x}{(x^2+a^2)^{n-1}} - a^2 \ \int \frac {x^{m-2} d x}{(x^2 + a^2)^n} $ | |
| $ \int \frac {d x}{x^m (x^2 +a^2)^n} = \frac {1}{a^2} \int \frac {d x}{x^m(x^2+a^2)^{n-1}} - \frac {1}{a^2} \int \frac {d x}{ x^{m-2}(x^2+a^2)^n} $ | |
| $ \text {9 Integeral Componant } x^2 - a^2 , x^2 > a^2 $ | |
| $ \int \frac {d x}{x^2 - a^2} = \frac {1}{2a} \ln \left ( \frac {x-a}{x+a} \right ) \text{ or } -\frac{1}{a} \operatorname{argcoth}\,\frac {x}{a} $ | |
| $ \int \frac {x d x}{x^2 - a^2} = \frac {1}{2} \ln(x^2 - a^2) $ | |
| $ \int \frac {x^2 d x}{x^2 -a^2} = x + \frac{a}{2} \ln \left ( \frac {x-a}{x+a} \right ) $ | |
| $ \int \frac {x^3 d x}{x^2 - a^2} = \frac{x^2}{2} + \frac {a^2}{2} \ln (x^2-a^2) $ | |
| $ \int \frac {d x}{x(x^2-a^2)} = \frac {1}{2a^2} \ln \left ( \frac {x^2 -a^2}{x^2} \right ) $ | |
| $ \int \frac {d x}{x^2(x^2-a^2)} = \frac {1}{a^2x} + \frac {1}{2a^3} \ln \left ( \frac {x-a}{x+a} \right ) $ | |
| $ \int \frac {d x}{x^3(x^2-a^2)} = \frac {1}{2a^2x^2} - \frac {1}{2a^4} \ln \left ( \frac {x^2}{x^2-a^2} \right ) $ | |
| $ \int \frac {d x}{(x^2 -a^2)^2} = \frac {-x}{2a^2(x^2-a^2)} - \frac {1}{4a^3} \ln \left ( \frac {x-a}{x+a} \right ) $ | |
| $ \int \frac {x d x}{(x^2 -a^2)^2} =\frac {-1}{2(x^2-a^2} $ | |
| $ \int \frac {x^2 d x}{(x^2 -a^2)^2} =\frac {-x}{2(x^2-a^2)} + \frac {1}{4a} \ln \left ( \frac {x-a}{x+a} \right ) $ | |
| $ \int \frac {x^3 d x}{(x^2 -a^2)^2} = \frac {-a^2}{2(x^2-a^2)} + \frac{1}{2} \ln (x^2-a^2) $ | |
| $ \int \frac {d x}{x(x^2 -a^2)^2} = \frac {-1}{2a^2(x^2-a^2)} + \frac {1}{2a^4} \ln \left ( \frac {x^2}{x^2-a^2} \right ) $ | |
| $ \int \frac {d x}{x^2(x^2 -a^2)^2} = - \frac {1}{a^4x} - \frac {x}{2a^4(x^2-a^2)} - \frac{3}{4a^5} \ln \left ( \frac {x-a}{x+a} \right ) $ | |
| $ \int \frac {d x}{x^3(x^2 -a^2)^2} = - \frac {1}{2a^4x^2} - \frac {1}{2a^4(x^2-a^2)} - \frac{1}{a^6} \ln \left ( \frac {x^2}{x^2-a^2} \right ) $ | |
| $ \int \frac {d x}{(x^2 -a^2)^n} = \frac {-x}{2(n-1)a^2(x^2-a^2)^{n-1}} - \frac {2n - 3}{(2n-2)a^2} \int \frac {d x}{(x^2-a^2)^{n-1}} $ | |
| $ \int \frac {x d x}{(x^2 -a^2)^n} = \frac {-1}{2(n-1)(x^2-a^2)^{n-1}} $ | |
| $ \int \frac {d x}{x(x^2 -a^2)^n} = \frac {-1}{2(n-1)a^2(x^2 - a^2)^{n-1}} - \frac {1}{a^2} \int \frac {d x}{x(x^2 -a^2)^{n-1}} $ | |
| $ \int \frac {x^m d x}{(x^2 -a^2)^n} = \int \frac {x^{m-2} d x}{(x^2 -a^2)^{n-1}} + a^2 \ \int \frac {x^{m-2} d x}{(x^2 -a^2)^n} $ | |
| $ \int \frac {d x}{x^m (x^2 -a^2)^n} =\frac {1}{a^2} \int \frac {d x}{x^{m-2} (x^2 -a^2)^n} - \frac{1}{a^2} \ \int \frac {d x}{x^m (x^2 -a^2)^{n-1}} $ | |
| 19 Integrals Component sin ax | |
| $ \int \sin a x d x = - \frac {\cos a x }{a} $ | |
| $ \int x \sin a x d x = \frac {\sin a x}{a^2}- \frac{x \cos a x}{a} $ | |
| $ \int x^2 \sin a x d x = \frac {2 x}{a^2} \sin a x + \left ( \frac {2}{a^3} - \frac {x^2}{a} \right)\cos a x $ | |
| $ \int x^3 \sin a x d x = \left( \frac {3 x^2}{a^2} - \frac{6}{a^4}\right)\sin a x + \left ( \frac {6x}{a^3} - \frac {x^3}{a} \right)\cos a x $ | |
| $ \int \frac {\sin a x}{x} d x = a x - \frac {(a x)^3}{3 \cdot 3!} + \frac {(a x)^5}{5 \cdot 5!} - \cdot \cdot \cdot $ | |
| $ \int \frac {\sin a x}{x^2} d x = - \frac {\sin a x}{x} + a \int \frac {\cos a x}{x}dx \qquad \left( \text {Voir} \text{ } 14.373 \right) $ | |
| $ \int \frac {d x}{\sin a x} = \frac {1}{a} \ln \left( \frac {1}{\sin a x} - \cot a x \right) = \frac {1}{a} \ln {\tan \frac{a x}{2}} $ | |
| $ \int \frac {x d x}{\sin a x} = \frac {1}{a^2} \left \{a x + \frac{(ax)^3}{18}+ \frac {7(ax)^5}{1800} + \cdot \cdot \cdot + \frac {2(2^{2n-1}-1)Bn(ax)^{2n+1}}{(2n+1)!} + \cdot\cdot\cdot \right \} $ | |
| $ \int \sin ^2 a x d x = \frac {x}{2}- \frac{\sin 2 a x}{4a} $ | |
| $ \int x \sin ^2 a x d x = \frac {x^2}{4}- \frac{x \sin 2 a x}{4a} - \frac {\cos 2 a x}{8a^2} $ | |
| $ \int \sin ^3 a x d x = -\frac {\cos a x}{a}- \frac{\cos^3 a x}{3a} $ | |
| $ \int \sin ^4 a x d x = \frac {3x}{8}- \frac{\sin 2 a x}{4a} + \frac {\sin4ax}{32a} $ | |
| $ \int \frac {d x}{\sin^2 a x} = -\frac {1}{a} \cot a x $ | |
| $ \int \frac {d x}{\sin^3 a x} = -\frac {\cos ax}{2a \sin^2 ax} + \frac{1}{2a}\ln \tan \frac{ax}{2} $ | |
| $ \int \sin px \sin q x d x = \frac {\sin (p-q)x}{2(p-q)} - \frac{\sin(p+q)x}{2(p+q)} \qquad \left( \text {Si} \text{ } p=\pm q, \text {voir}\text{ }14.368 \right) $ | |
| $ \int \frac {d x}{1-\sin a x} = \frac {1}{a} \tan {\left ( \frac{\pi}{4}+\frac{ax}{2}\right )} $ | |
| $ \int \frac {x d x}{1-\sin a x} = \frac {x}{a} \tan {\left ( \frac{\pi}{4}+\frac{ax}{2}\right )} +\frac{2}{a^2}\ln\sin{\left ( \frac{\pi}{4}-\frac{ax}{2}\right )} $ | |
| $ \int \frac {d x}{1+\sin a x} = -\frac {1}{a} \tan {\left ( \frac{\pi}{4}-\frac{ax}{2}\right )} $ | |
| $ \int \frac {x d x}{1+\sin a x} = -\frac {x}{a} \tan {\left ( \frac{\pi}{4}-\frac{ax}{2}\right )} +\frac{2}{a^2}\ln\sin{\left ( \frac{\pi}{4}+\frac{ax}{2}\right )} $ | |
| $ \int \frac {d x}{(1-\sin a x)^2} = \frac {1}{2a} \tan {\left ( \frac{\pi}{4}+\frac{ax}{2}\right )} +\frac{1}{6a}\tan^3{\left ( \frac{\pi}{4}+\frac{ax}{2}\right )} $ | |
| $ \int \frac {d x}{(1+\sin a x)^2} = -\frac {1}{2a} \tan {\left ( \frac{\pi}{4}-\frac{ax}{2}\right )} -\frac{1}{6a}\tan^3{\left ( \frac{\pi}{4}-\frac{ax}{2}\right )} $ | |
| $ \int \frac {dx}{(p+q \sin ax)}= \begin{cases} \frac{2}{a \sqrt {p^2-q^2}} \arctan \frac {p\tan \frac{1}{2}ax + q}{\sqrt{p^2-q^2}} \\ \frac{1}{a\sqrt{q^2-p^2}} \ln \left( \frac{p \tan \frac{1}{2} ax + q-\sqrt{q^2-p^2}}{p \tan \frac{1}{2}ax+q+\sqrt{q^2-p^2}} \right) \end{cases} \qquad \left ( \text{Si }p= \pm q \text{ voir }14.354 \text{ et } 14.356 \right ) $ | |
| $ \int \frac {d x}{(p+q \sin a x)^2}=\frac{q \cos ax}{a(p^2-q^2)(p+q \sin ax)}+\frac{p}{p^2-q^2}\int\frac{dx}{p+q \sin ax} \qquad \left ( \text{Si }p= \pm q \text{ voir }14.358 \text{ et } 14.359 \right ) $ | |
| $ \int \frac {d x}{p^2+q^2 \sin^2 a x}= \frac{1}{a p \sqrt {p^2+q^2}} \arctan \frac {\sqrt {p^2+q^2} \tan ax}{p} $ | |
| $ \int \frac {dx}{(p^2-q^2 \sin^2 a x)}= \begin{cases} \frac{1}{a p \sqrt {p^2-q^2}} \arctan \frac {\sqrt {p^2-q^2} \tan ax }{p} \\ \frac{1}{2 a p \sqrt {q^2-p^2}} \ln \left( \frac{ \sqrt{q^2-p^2} \tan ax + p}{\sqrt{q^2-p^2} \tan a x -p} \right) \end{cases} $ | |
| $ \int x^m \sin a x d x = -\frac {x^m \cos ax}{a} + \frac{m x^{m-1} \sin ax }{a^2} - \frac {m(m-1)}{a^2} \int x^{m-2} \sin ax dx $ | |
| $ \int \frac {\sin a x}{x^n} d x = - \frac {\sin a x}{(n-1) x^{n-1}} + \frac {a}{n-1} \int \frac {\cos ax }{x^{n-1}}dx \qquad \left ( \text{ Voir }14.395 \right ) $ | |
| $ \int \sin^n a x d x = -\frac {\sin^{n-1}ax \cos ax}{an} + \frac{n-1}{n} \int \sin^{n-2} ax dx $ | |
| $ \int \frac {dx}{\sin^n a x}= \frac { -\cos a x}{a(n-1) \sin^{n-1}ax} + \frac {n-2}{n-1} \int \frac {dx }{\sin^{n-2}ax} $ | |
| $ \int \frac {x dx}{\sin^n a x}= \frac {- x \cos a x}{a(n-1) \sin^{n-1}ax} - \frac {1}{a^2(n-2)(n-1)\sin^{n-2}ax} +\frac {n-2}{n-1} \int \frac {x dx }{\sin^{n-2}ax} $ | |
| 20 Integrals Component cos ax | |
| $ \int \cos a x d x = \frac {\sin a x }{a} $ | |
| $ \int x \cos a x d x = \frac {\cos a x}{a^2} + \frac{x \sin a x}{a} $ | |
| $ \int x^2 \cos a x d x = \frac {2 x}{a^2} \cos a x + \left ( \frac {x^2}{a} - \frac {2}{a^3} \right)\sin a x $ | |
| $ \int x^3 \cos a x d x = \left( \frac {3 x^2}{a^2} - \frac{6}{a^4}\right)\cos a x + \left ( \frac {x^3}{a} - \frac{6x}{a^3} \right)\sin a x $ | |
| $ \int \frac {\cos a x}{x} d x = \ln x - \frac {(a x)^2}{2 \cdot 2!} + \frac {(a x)^4}{4 \cdot 4!} - \frac {(a x)^6}{6 \cdot 6!} \cdot \cdot \cdot $ | |
| $ \int \frac {\cos a x}{x^2} d x = - \frac {\cos a x}{x} - a \int \frac {\sin a x}{x}dx \qquad \left( \text {Voir} \text{ } 14.343 \right) $ | |
| $ \int \frac {d x}{\cos a x} = \frac {1}{a} \ln \left( \frac {1}{\cos a x} +\tan a x \right)= \frac {1}{a} \tan {\left ( \frac{\pi}{4}+\frac{ax}{2}\right )} $ | |
| $ \int \frac {x d x}{\cos a x} = \frac {1}{a^2} \left \{\frac{(a x)^2}{2} + \frac{(ax)^4}{8}+ \frac {5(ax)^6}{144} + \cdot \cdot \cdot + \frac {En(ax)^{2n+2}}{(2n+2)(2n)!} + \cdot\cdot\cdot \right \} $ | |
| $ \int \cos ^2 a x d x = \frac {x}{2}+ \frac{\sin 2 a x}{4a} $ | |
| $ \int x \cos ^2 a x d x = \frac {x^2}{4}+ \frac{x \sin 2 a x}{4a} + \frac {\cos 2 a x}{8a^2} $ | |
| $ \int \cos ^3 a x d x = \frac {\sin a x}{a}- \frac{\sin^3 a x}{3a} $ | |
| $ \int \cos ^4 a x d x = \frac {3x}{8}+ \frac{\sin 2 a x}{4a} + \frac {\sin4ax}{32a} $ | |
| $ \int \frac {d x}{\cos^2 a x} = \frac {\cot a x}{a} $ | |
| $ \int \frac {d x}{\cos^3 a x} = \frac {\sin ax}{2a \cos^2 ax} + \frac{1}{2a}\ln \tan { \left( \frac{\pi}{4}+ \frac {ax}{2} \right)} $ | |
| $ \int \cos ax \cos p x d x = \frac {\sin (a-p)x}{2(a-p)} - \frac{\sin(a+p)x}{2(a+p)} \qquad \left( \text {Si } a=\pm p, \text { voir } 14.377 \right) $ | |
| $ \int \frac {d x}{1-\cos a x} = -\frac {1}{a} \cot \frac{ax}{2} $ | |
| $ \int \frac {x d x}{1-\cos a x} = -\frac {x}{a} \cot \frac{ax}{2} +\frac{2}{a^2}\ln\sin \frac{ax}{2} $ | |
| $ \int \frac {d x}{1+\cos a x} = \frac {1}{a} \tan \frac{ax}{2} $ | |
| $ \int \frac {x d x}{1+\cos a x} = \frac {x}{a} \tan \frac{ax}{2} + \frac{2}{a^2}\ln\cos \frac{ax}{2} $ | |
| $ \int \frac {d x}{(1-\cos a x)^2} = -\frac {1}{2a} \cot \frac{ax}{2} - \frac{1}{6a}\cot^3 \frac{ax}{2} $ | |
| $ \int \frac {d x}{(1+\cos a x)^2} = \frac {1}{2a} \tan \frac{ax}{2} + \frac{1}{6a}\tan^3 \frac{ax}{2} $ | |
| $ \int \frac {dx}{(p+q \cos ax)}= \begin{cases} \frac{2}{a \sqrt {p^2-q^2}} \arctan \sqrt {(p-q)/(p+q)} \tan \frac{1}{2}ax \\ \frac{1}{a \sqrt{q^2-p^2}} \ln \left( \frac{ \tan \frac{1}{2} ax + \sqrt{(q+p)/(q-p)}}{\tan \frac{1}{2}ax-\sqrt{(q+p)/(q-p)}} \right) \end{cases} \qquad \left ( \text{Si }p= \pm q \text{ voir }14.384 \text{ et } 14.386 \right ) $ | |
| $ \int \frac {d x}{(p+q \cos a x)^2}=\frac{q \sin ax}{a(q^2-p^2)(p+q \cos ax)}-\frac{p}{q^2-p^2}\int\frac{dx}{p+q \cos ax} \qquad \left ( \text{Si }p= \pm q \text{ voir }14.388 \text{ et } 14.389 \right ) $ | |
| $ \int \frac {d x}{p^2+q^2 \cos^2 a x}= \frac{1}{a p \sqrt {p^2+q^2}} \arctan \frac {p \tan ax}{\sqrt {p^2+q^2} } $ | |
| $ \int \frac {dx}{(p^2-q^2 \cos^2 a x)}= \begin{cases} \frac{1}{a p \sqrt {p^2-q^2}} \arctan \frac {p \tan ax }{\sqrt {p^2-q^2} } \\ \frac{1}{2 a p \sqrt {q^2-p^2}} \ln \left( \frac{ p \tan ax - \sqrt{q^2-p^2}}{p \tan a x + \sqrt{q^2-p^2}} \right) \end{cases} $ | |
| $ \int x^m \cos a x d x = -\frac {x^m \sin ax}{a} + \frac{m x^{m-1}}{a^2} \sin ax - \frac {m(m-1)}{a^2} \int x^{m-2} \cos ax dx $ | |
| $ \int \frac {\cos a x}{x^n} d x = - \frac {\cos a x}{(n-1) x^{n-1}} - \frac {a}{n-1} \int \frac {\sin ax }{x^{n-1}}dx \qquad \left ( \text{ Voir }14.365 \right ) $ | |
| $ \int \cos^n a x d x = -\frac {\sin ax \cos^{n-1}ax }{an} + \frac{n-1}{n} \int \cos^{n-2} ax dx $ | |
| $ \int \frac {dx}{\cos^n a x}= \frac { \sin a x \cos^{n-1}ax}{an} + \frac {n-2}{n-1} \int \frac {dx }{\cos^{n-2}ax} $ | |
| $ \int \frac {x dx}{\cos^n a x}= \frac {x \sin a x}{a(n-1) \cos^{n-1}ax} - \frac {1}{a^2(n-1)(n-2)\cos^{n-2}ax} +\frac {n-2}{n-1} \int \frac {x dx }{\cos^{n-2}ax} $ | |
| 21 Integrals Component sin ax ET cos ax | |
| $ \int \sin ax \cos a x d x = \frac {\sin^2 a x }{2a} $ | |
| $ \int \sin p x \cos q x d x = -\frac {\cos (p-q)x}{2(p-q)} - \frac{\cos(p+q)x}{2(p+q)} $ | |
| $ \int \sin^n x \cos a x d x = \frac{ \sin^{n+1} ax}{(n+1)a} \qquad \left ( \text{Si }n=-1, \text{ voir }14.440\right ) $ | |
| $ \int \cos^n x \sin a x d x = -\frac{ \cos^{n+1} ax}{(n+1)a} \qquad \left ( \text{Si }n=-1, \text{ voir }14.429 \right ) $ | |
| $ \int \sin^2 a x \cos^2 a x d x = \frac{x}{8} - \frac {\sin 4ax}{32a} $ | |
| $ \int \frac {d x}{\sin ax \cos a x} = \frac {1}{a} \tan ax $ | |
| $ \int \frac {d x}{\sin^2 ax \cos a x} = \frac {1}{a} \ln \tan { \left( \frac{\pi}{4}+ \frac {ax}{2} \right)} - \frac {1}{a \sin ax } $ | |
| $ \int \frac {d x}{\sin ax \cos^2 a x} = \frac {1}{a} \ln \tan \frac {ax}{2} + \frac {1}{a \cos ax } $ | |
| $ \int \frac {d x}{\sin^2 ax \cos^2 a x} = -\frac {2\cot 2 a x }{a} $ | |
| $ \int \frac {\sin^2 ax }{\cos a x}dx = -\frac {\sin a x}{a} +\frac {1}{a} \ln \tan { \left( \frac {ax}{2}+ \frac{\pi}{4} \right)} $ | |
| $ \int \frac {\cos^2 ax }{\sin a x}dx = \frac {\cos ax}{a} + \frac{1}{a} \ln \tan \frac{ax}{2} $ | |
| $ \int \frac {dx}{\cos ax (1 \pm \sin a x} = \mp \frac {1}{2a (1 \pm \sin ax} + \frac{1}{2a} \ln \tan { \left( \frac {ax}{2}+ \frac{\pi}{4} \right)} $ | |
| $ \int \frac {dx}{\sin ax (1 \pm \cos a x} = \pm \frac {1}{2a (1 \pm \cos ax} + \frac{1}{2a} \ln \tan \frac {ax}{2} $ | |
| $ \int \frac {dx}{\sin ax \pm \cos a x} = \frac {1}{a \sqrt {2}} \ln \tan { \left( \frac {ax}{2} \pm \frac{\pi}{8} \right)} $ | |
| $ \int \frac {\sin ax dx}{\sin ax \pm \cos a x} = \frac {x}{2} \mp \frac {1}{2a} \ln { \left( \sin ax \pm \cos ax \right)} $ | |
| $ \int \frac {\cos ax dx}{\sin ax \pm \cos a x} = \mp \frac {x}{2} + \frac {1}{2a} \ln { \left( \sin ax \pm \cos ax \right)} $ | |
| $ \int \frac {\sin ax dx}{p+q \cos a x} = -\frac {1}{aq} \ln { \left( p+ q\cos ax \right)} $ | |
| $ \int \frac {\cos ax dx}{p+q \cos a x} = \frac {1}{aq} \ln { \left( p+ q\sin ax \right)} $ | |
| $ \int \frac {\sin ax dx}{(p+q \cos a x)^n} = \frac {1}{aq(n-1)(p+q \cos ax)^{n-1}} $ | |
| $ \int \frac {\sin ax dx}{(p+q \sin a x)^n} = \frac {-1}{aq(n-1)(p+q \sin ax)^{n-1}} $ | |
| $ \int \frac {dx}{p \sin ax + q \cos a x} = \frac {1}{a \sqrt {p^2+q^2}} \ln \tan { \left( \frac {ax+\arctan (q/p)}{2} \right)} $ | |
| $ \int \frac {dx}{p \sin ax +q \cos ax + r}= \begin{cases} \frac{2}{a \sqrt {r^2 -p^2-q^2}} \arctan \left( \frac {p+(r-q) \tan (ax/2)}{\sqrt {r^2 - p^2-q^2}} \right) \\ \frac{1}{a \sqrt {q^2 +p^2-r^2}} \ln \left( \frac{ p- \sqrt{p^2+q^2-r^2}+(r+q)\tan \frac{ax}{2} }{p+\sqrt{p^2+q^2-r^2}+(r+q)\tan \frac{ax}{2}} \right) \end{cases} \left ( \text{ Si }r=q \text{ voir }14.421. \text{ Si } r^2=p^2+q^2 \text{ voir }14.422 \right ) $ | |
| $ \int \frac {dx}{p \sin ax +q (1+\cos ax )}= \frac{1}{ap} \ln \left( q+ p \tan \frac{ax}{2} \right) $ | |
| $ \int \frac {dx}{p \sin ax +q \cos ax \pm \sqrt {p^2+q^2}}= \frac {-1}{a \sqrt {p^2+q^2}} \tan { \left( \frac{\pi}{4} \mp \frac{ax+ \arctan{(q/p)}}{2} \right)} $ | |
| $ \int \frac {dx}{p^2 \sin^2 ax +q^2 \cos^2 ax }= \frac{1}{apq} \arctan { \frac {p \tan ax}{q}} $ | |
| $ \int \frac {dx}{p^2 \sin^2 ax - q^2 \cos^2 ax }= \frac{1}{2apq} \ln { \left( \frac {p \tan ax -q}{p \tan ax + q} \right)} $ | |
| $ \int \sin^m ax \cos^n ax dx= \begin{cases} -\frac{\sin^{m-1} ax \cos^{n-1}ax}{a (m+n)} + \frac {m-1}{m+n} \int \sin^{m-2} ax \cos^n ax dx \\ \frac{\sin^{m-1} ax \cos^{n+1}ax}{a (m+n)} + \frac{ n-1 }{m+n} \int \sin^m ax \cos^{n-2} ax dx \end{cases} $ | |
| $ \int \frac {\sin^m ax}{ \cos^n ax }dx= \begin{cases} \frac{\sin^{m-1} ax } {a(n-1) \cos^{n-1}ax} -\frac {m-1}{n-1} \int \sin^{m-2} ax \cos^{n-2} ax dx \\ \frac{\sin^{m+1} ax } {a(n-1) \cos^{n+1}ax} - \frac{m-n+2}{n-1} \int \sin^m ax \cos^{n-2} ax dx \\ \frac{- \sin^{m-1} ax } {a(m-n) \cos^{n-1}ax} + \frac {m-1}{m-n} \int \sin^{m-2} ax \cos^n ax dx \\ \end{cases} $ | |
| $ \int \frac {\cos^m ax}{ \sin^n ax }dx= \begin{cases} \frac{-\cos^{m-1} ax } {a(n-1) \sin^{n-1}ax} -\frac {m-1}{n-1} \int \cos^{m-2} ax \sin^{n-2} ax dx \\ \frac{-\cos^{m+1} ax } {a(n-1) \sin^{n-1}ax} - \frac{m-n+2}{n-1} \int \cos^m ax \sin^{n-2} ax dx \\ \frac{- \cos^{m-1} ax } {a(m-n) \sin^{n-1}ax} + \frac {m-1}{m-n} \int \cos^{m-2} ax \sin^n ax dx \\ \end{cases} $ | |
| $ \int \frac {dx}{ \sin^m ax \cos^n a x} \begin{cases} \frac{1}{a(n-1) \sin^{n-1}ax \cos^{n-1} ax} + \frac{m-n+2}{n-1} \frac {dx}{ \sin^m ax \cos^{n-2}} \\ \frac{-1}{a(n-1) \sin^{m-1}ax \cos^{n-1} ax} + \frac{m-n+2}{n-1} \frac {dx}{ \sin^{m-2} ax \cos^n ax}\\ \end{cases} $ | |
| 22 Integrals Component tan ax | |
| $ \int \tan a x d x = - \frac {1}{a} \ln {\cos a x } $ | |
| $ \int \tan ^2 a x d x = \frac { \tan ax}{a} - x $ | |
| $ \int \tan ^3 a x d x = \frac {\tan^2 ax}{2a}+ \frac{1}{a} \ln {\cos a x} $ | |
| $ \int \frac {\tan^n ax }{\cos^2 a x}dx = \frac {\tan^{n+1} a x}{(n+1)a} $ | |
| $ \int \frac {1}{\cos^2 a x \tan ax }dx = \frac {1}{a} \ln {\tan a x} $ | |
| $ \int \frac {dx}{ \tan ax } = \frac {1}{a} \ln {\sin a x} $ | |
| $ \int x \tan ax dx = \frac {1}{a^2} \left \{\frac{(a x)^3}{3} + \frac{(ax)^5}{15}+ \frac {2(ax)^7}{105} + \cdot \cdot \cdot + \frac {2^{2n}(2^{2n-1})Bn(ax)^{2n-1}}{(2n+1)!} + \cdot \cdot \cdot \right \} $ | |
| $ \int \frac {\tan ax }{ x } dx = ax + \frac{(a x)^3}{9} + \frac{2(ax)^5}{75} + \cdot \cdot \cdot + \frac {2^{2n}(2^{2n-1})Bn(ax)^{2n-1}}{(2n-1)(2n)!} + \cdot \cdot \cdot $ | |
| $ \int x \tan^2 ax dx = \frac {x \tan ax}{a} + \frac {1}{a^2} \ln {\cos a x} - \frac {x^2}{2} $ | |
| $ \int \frac {dx}{p+q \tan ax} = \frac {px}{p^2+q^2} + \frac {q}{a(p^2+q^2)} \ln {\left( q\sin a x + p \cos ax \right)} $ | |
| $ \int \tan^n ax dx = \frac {\tan^{n+1}ax}{(n+1)a} -\int \tan^{n-2} a x dx $ | |
| 23 Integrals Component cot ax | |
| $ \int \cot a x d x = \frac {1}{a} \ln {\sin a x } $ | |
| $ \int \cot ^2 a x d x = -\frac { \cot ax}{a} - x $ | |
| $ \int \cot ^3 a x d x = - \frac {\cot^2 ax}{2a} - \frac{1}{a} \ln {\sin a x} $ | |
| $ \int \frac {\cot^n ax }{\sin^2 a x}dx = -\frac {\cot^{n+1} a x}{(n+1)a} $ | |
| $ \int \frac {dx}{\sin^2 a x \cot ax }= - \frac {1}{a} \ln {\cot a x} $ | |
| $ \int \frac {dx}{ \cot ax } = -\frac {1}{a} \ln {\cos a x} $ | |
| $ \int x \cot ax dx = \frac {1}{a^2} \left \{ ax - \frac{(a x)^3}{9} - \cdot \cdot \cdot - \frac {2^{2n}Bn(ax)^{2n+1}}{(2n+1)!} + \cdot \cdot \cdot \right \} $ | |
| $ \int \frac {\cot ax } {x} dx = -\frac {1}{ax} - \frac{a x}{3} - \frac{(ax)^3}{135} - \cdot \cdot \cdot - \frac {2^{2n}Bn(ax)^{2n-1}}{(2n-1)(2n)!} - \cdot \cdot \cdot $ | |
| $ \int x \cot^2 ax dx = - \frac {x \cot ax}{a} + \frac {1}{a^2} \ln {\sin a x} - \frac {x^2}{2} $ | |
| $ \int \frac {dx}{p+q \cot ax} = \frac {px}{p^2+q^2} - \frac {q}{a(p^2+q^2)} \ln {\left( p\sin a x + q \cos ax \right)} $ | |
| $ \int \cot^n ax dx = -\frac {\cot^{n-1}ax}{(n+1)a} -\int \cot^{n-2} a x dx $ | |
| 24 Integrals Component $ \frac {1}{\cos ax} $ | |
| $ \int \frac {dx }{\cos a x}dx = \frac {1}{a} \ln { \left( \frac {1} {\cos ax} + \tan ax \right ) } = \frac {1}{a} \ln { \left( \frac {ax}{2} +\frac {\pi}{4} \right) } $ | |
| $ \int \frac {dx}{ \cos^2 ax }= \frac {\tan ax} {a} $ | |
| $ \int \frac {1}{ \cos^3 ax }dx = \frac {\tan ax}{2a \cos ax}+ \frac {1}{2a} \ln { \left( \frac{1}{\cos ax}+ {\tan a x} \right) } $ | |
| $ \int \frac {1}{\cos^n ax} \tan ax dx= \frac{a}{na \cos^nax} $ | |
| $ \int \cos ax dx = \frac {\sin ax}{a} $ | |
| $ \int \frac {xdx} {\cos ax} = \frac {1}{a^2} \left \{ \frac {(ax)^2}{2}+ \frac{(a x)^4}{8}+\frac {5(ax)^6}{144} + \cdot \cdot \cdot + \frac {En(ax)^{2n+2}}{(2n+2)(2n)!} + \cdot \cdot \cdot \right \} $ | |
| $ \int \frac {dx}{x \cos ax } = \ln {x} + \frac {(ax)^2} {4} + \frac{5(a x)^4}{96}+\frac{61(ax)^6 }{4320}+ \cdot \cdot \cdot + \frac {En(ax)^{2n}}{2n(2n)!} + \cdot \cdot \cdot $ | |
| $ \int x \cos^2 ax dx = - \frac {x \cot ax}{a} + \frac {1}{a^2} \ln {\sin a x} - \frac {x^2}{2} $ | |
| $ \int \frac {x dx}{ \cos^2 ax} = \frac {x}{a} \tan ax + \frac {1}{a^2} \ln { \cos ax } $ | |
| $ \int \frac {dx}{q+\frac {p}{\cos ax}}= \frac{x}{q}+\frac{p}{q} \int \frac{dx}{p+q\cos ax} $ | |
| $ \int \cos^n ax dx = \frac {\tan ax}{a(n-1)\cos^{n-2}ax} + \frac {n-2}{n-1} \int \frac {dx}{\cos^{n-2} a x } $ | |
| 25 Integrals Component $ \frac {1}{\sin ax} $ | |
| $ \int \frac {dx }{\sin a x}dx = \frac {1}{a} \ln { \left( \frac {1} {\sin ax} - \cot ax \right ) } = \frac {1}{a} \ln { \tan \frac {ax}{2} } $ | |
| $ \int \frac {dx}{ \sin^2 ax }= -\frac {\cot ax} { a} $ | |
| $ \int \frac {1}{ \sin^3 ax }dx = -\frac {\cot ax}{2a \sin ax}+ \frac {1}{2a} \ln{ \tan \frac{ax}{2} } $ | |
| $ \int \frac {\cot ax dx}{\sin^n ax}= -\frac{1}{na \sin^nax} $ | |
| $ \int \sin ax dx = -\frac {\cos ax}{a} $ | |
| $ \int \frac {xdx} {\sin ax} = \frac {1}{a^2} \left \{ ax + \frac {(ax)^3}{18}+ \frac{7(a x)^5}{1800} + \cdot \cdot \cdot + \frac {2(2^{2n-1}-1)Bn(ax)^{2n+1}}{(2n+2)!} + \cdot \cdot \cdot \right \} $ | |
| $ \int \frac {dx}{x \sin ax } = -\frac{1}{ax} + \frac {ax} {6} + \frac{7(a x)^3}{1080}+ \cdot \cdot \cdot + \frac {2(2^{2n-1}-1)Bn(ax)^{2n+1}}{(2n-1)(2n)!} + \cdot \cdot \cdot $ | |
| $ \int x \sin^2 ax dx = - \frac {x \cot ax}{a} + \frac {1}{a^2} \ln {\sin a x} - \frac {x^2}{2} $ | |
| $ \int \frac {x dx}{ \cos^2 ax} = -\frac {x\cot ax}{a} + \frac {1}{a^2} \ln { \sin ax } $ | |
| $ \int \frac {dx}{q+\frac {p}{\sin ax}}= \frac{x}{q}-\frac{p}{q} \int \frac{dx}{p+q\sin ax} $ | |
| $ \int \sin^n ax dx = -\frac {\cot ax}{a(n-1)\sin^{n-2}ax} + \frac {n-2}{n-1} \int \frac {dx}{\sin^{n-2} a x } $ | |
| 26 Integrals Component the inverse circular functions | |
| $ \int \arcsin \frac {x }{ a}dx = x \arcsin \frac {x}{a} + \sqrt { a^2-x^2 } $ | |
| $ \int x \arcsin \frac {x }{ a}dx = \left ( \frac{x^2}{2}-\frac {a^2}{4} \right) \arcsin \frac {x}{a} + \frac {x \sqrt { a^2-x^2 }}{4} $ | |
| $ \int x^2 \arcsin \frac {x }{ a}dx = \frac{x^3}{3}\arcsin \frac {x}{a} + \frac {\left( x^2+2a^2 \right) \sqrt { a^2-x^2 }}{9} $ | |
| $ \int \frac {\arcsin (x/a)}{x}dx = \frac {x}{a} + \frac {(a/x)^3}{2 \cdot 3 \cdot 3}+ \frac{1 \cdot 3(x/a)^5}{2 \cdot 4 \cdot 5 \cdot 5} + \frac {1 \cdot 3 \cdot 5 (x/a)^7}{2 \cdot 4 \cdot 6 \cdot 7 \cdot 7} + \cdot \cdot \cdot $ | |
| $ \int \frac {\arcsin (x/a)}{x^2}dx = -\frac{\arcsin (x/a)}{x} - \frac {1}{a} \ln { \left( \frac {a +\sqrt{a^2-x^2}}{x} \right)} $ | |
| $ \int \left( \arcsin \frac{x}{a} \right)^2 dx = x \left( \arcsin \frac{x}{a} \right)^2-2x + 2 \sqrt{a^2-x^2} \arcsin \frac{x}{a} $ | |
| $ \int \arccos \frac {x }{ a}dx = x \arccos \frac {x}{a} - \sqrt { a^2-x^2 } $ | |
| $ \int x \arccos \frac {x }{ a}dx = \frac{x^2}{2}-\frac {a^2}{4} \arccos \frac {x}{a} - \frac {x \sqrt { a^2-x^2 }}{4} $ | |
| $ \int x^2 \arccos \frac {x }{ a}dx = \frac{x^3}{3}\arccos \frac {x}{a} - \frac {\left( x^2+2a^2 \right) \sqrt { a^2-x^2 }}{9} $ | |
| $ \int \frac {\arccos (x/a)}{x}dx = \frac {\pi}{2}\ln{x} - \int \frac {\arcsin (x/a)}{x} dx \qquad \left( \text {voir } 14.474 \right) $ | |
| $ \int \frac {\arccos (x/a)}{x^2}dx = -\frac{\arccos (x/a)}{x} + \frac {1}{a} \ln { \left( \frac {a +\sqrt{a^2-x^2}}{x} \right)} $ | |
| $ \int \left( \arccos \frac{x}{a} \right)^2 dx = x \left( \arccos \frac{x}{a} \right)^2-2x - 2 \sqrt{a^2-x^2} \arccos \frac{x}{a} $ | |
| $ \int \arctan \frac {x }{ a}dx = x \arctan \frac {x}{a} - \frac {a}{2} \ln {x^2+a^2} $ | |
| $ \int x \arctan \frac {x }{ a}dx = \frac{1}{2}(x^2+a^2)\arctan \frac {x}{a} - \frac {ax}{2} $ | |
| $ \int x^2 \arctan \frac {x }{ a}dx = \frac {x^3}{3}\arctan \frac{x}{a} - \frac {ax^2}{6} +\frac{a^3}{6} \ln \left (x^2+a^2) \right) $ | |
| $ \int \frac {\arctan (x/a)}{x}dx = \frac {x}{a} - \frac{(x/a)^3}{3^2} + \frac {(x/a)^5}{5^2} - \frac {(x/a)^7)}{7^2} + \cdot \cdot \cdot $ | |
| $ \int \frac {\arctan (x/a)}{x^2}dx = -\frac{1}{x}\arctan \frac{x}{a} - \frac {a}{2} \ln { \left( \frac {\sqrt{a^2+x^2}}{x^2} \right)} $ | |
| $ \int \arccot \frac {x }{ a}dx = x \arccot \frac {x}{a} + \frac {a}{2} \ln {x^2+a^2} $ | |
| $ \int x \arccot \frac {x }{ a}dx = \frac{1}{2}(x^2+a^2)\arccot \frac {x}{a} + \frac {ax}{2} $ | |
| $ \int x^2 \arccot \frac {x }{ a}dx = \frac {x^3}{3}\arccot \frac{x}{a} + \frac {ax^2}{6} - \frac{a^3}{6} \ln \left (x^2+a^2) \right) $ | |
| $ \int \frac {\arccot (x/a)}{x}dx = \frac {\pi}{2}\ln{x} - \int \frac{\arctan(x/a)}{x}dx $ | |
| $ \int \frac {\arccot (x/a)}{x^2}dx = -\frac{\arccot(x/a)}{x} + \frac {1}{2a} \ln { \left( \frac {a^2+x^2}{x^2} \right)} $ | |
| $ \int \arccos \frac {a}{x}dx= \begin{cases} x \arccos \frac{a}{x} - a \ln \left( x + \sqrt {x^2-a^2} \right) \qquad 0<\arccos \frac{a}{x}< \frac{\pi}{2} \\ x \arccos \frac{a}{x} + a \ln \left( x + \sqrt {x^2-a^2} \right) \qquad \frac {\pi}{2}<\arccos \frac{a}{x}< \pi \\ \end{cases} $ | |
| $ \int x \arccos \frac {a}{x}dx= \begin{cases} \frac {x^2}{2} \arccos \frac{a}{x} - \frac {a \sqrt {x^2-a^2}}{2} \qquad 0<\arccos \frac{a}{x}< \frac{\pi}{2} \\ \frac {x^2}{2} \arccos \frac{a}{x} + \frac {a \sqrt {x^2-a^2}}{2} \qquad \frac {\pi}{2}<\arccos \frac{a}{x}< \pi \\ \end{cases} $ | |
| $ \int x^2 \arccos \frac {a}{x}dx= \begin{cases} \frac {x^3}{3} \arccos \frac{a}{x} - \frac {ax \sqrt {x^2-a^2}}{6} - \frac {a^3}{6} \ln {\left(x+\sqrt {x^2-a^2}\right) }\qquad 0<\arccos \frac{a}{x}< \frac{\pi}{2} \\ \frac {x^3}{3} \arccos \frac{a}{x} + \frac {ax \sqrt {x^2-a^2}}{6} - \frac {a^3}{6} \ln {\left(x+\sqrt {x^2-a^2}\right) }\qquad \frac {\pi}{2}<\arccos \frac{a}{x}< \pi \\ \end{cases} $ | |
| $ \int \frac {\arccos (x/a)}{x}dx = \frac {\pi}{2} + \frac {a}{x}+ \frac {(a/x)^3}{2 \cdot 3 \cdot 3}+ \frac{1 \cdot 3(x/a)^5}{2 \cdot 4 \cdot 5 \cdot 5} + \frac {1 \cdot 3 \cdot 5 (x/a)^7}{2 \cdot 4 \cdot 6 \cdot 7 \cdot 7} + \cdot \cdot \cdot $ | |
| $ \int \frac {\arccos (x/a)}{x^2} dx= \begin{cases} -\frac{\arccos(a/x)}{x}+\frac{\sqrt{x^2-a^2}}{ax} \qquad 0<\arccos \frac{a}{x}< \frac{\pi}{2} \\ -\frac{\arccos(a/x)}{x}-\frac{\sqrt{x^2-a^2}}{ax} \qquad \frac{\pi}{2}< \arccos \frac{a}{x}<{\pi} \\ \end{cases} $ | |
| $ \int \arcsin \frac {a}{x}dx= \begin{cases} x \arcsin \frac{a}{x} + a \ln \left( x + \sqrt {x^2-a^2} \right) \qquad 0<\arcsin \frac{a}{x}< \frac{\pi}{2} \\ x \arcsin \frac{a}{x} + a \ln \left( x - \sqrt {x^2-a^2} \right) \qquad -\frac {\pi}{2}<\arcsin \frac{a}{x}< 0 \\ \end{cases} $ | |
| $ \int x \arcsin \frac {a}{x}dx= \begin{cases} \frac {x^2}{2} \arcsin \frac{a}{x} + \frac {a \sqrt {x^2-a^2}}{2} \qquad 0<\arccos \frac{a}{x}< \frac{\pi}{2} \\ \frac {x^2}{2} \arcsin \frac{a}{x} + \frac {a \sqrt {x^2-a^2}}{2} \qquad -\frac {\pi}{2}<\arccos \frac{a}{x}< 0 \\ \end{cases} $ | |
| $ \int x^2 \arcsin \frac {a}{x}dx= \begin{cases} \frac {x^3}{3} \arcsin \frac{a}{x} + \frac {a \sqrt {x^2-a^2}}{2} + \frac {a^3}{6} \ln {\left(x+\sqrt {x^2-a^2}\right) }\qquad 0<\arcsin \frac{a}{x}< \frac{\pi}{2} \\ \frac {x^3}{3} \arcsin \frac{a}{x} - \frac {a \sqrt {x^2-a^2}}{2} - \frac {a^3}{6} \ln {\left(x+\sqrt {x^2-a^2}\right) }\qquad -\frac {\pi}{2}<\arccos \frac{a}{x}< 0 \\ \end{cases} $ | |
| $ \int \frac {\arcsin (x/a)}{x}dx = - \left( \frac {a}{x}+ \frac {(a/x)^3}{2 \cdot 3 \cdot 3}+ \frac{1 \cdot 3(x/a)^5}{2 \cdot 4 \cdot 5 \cdot 5} + \frac {1 \cdot 3 \cdot 5 (x/a)^7}{2 \cdot 4 \cdot 6 \cdot 7 \cdot 7} + \cdot \cdot \cdot \right) $ | |
| $ \int \frac {\arcsin (x/a)}{x^2} dx= \begin{cases} -\frac{\arcsin(a/x)}{x}-\frac{\sqrt{x^2-a^2}}{ax} \qquad 0<\arcsin \frac{a}{x}< \frac{\pi}{2} \\ -\frac{\arcsin(a/x)}{x}+\frac{\sqrt{x^2-a^2}}{ax} \qquad -\frac{\pi}{2}< \arccos \frac{a}{x}<0 \\ \end{cases} $ | |
| $ \int x^m \arcsin \frac {x}{a}dx=\frac{x^{m+1}}{m+1}\arcsin \frac {x}{a} - \frac {1}{m+1} \int \frac{x^{m+1}}{\sqrt{a^2-x^2}}dx $ | |
| $ \int x^m \arccos \frac {x}{a}dx=\frac{x^{m+1}}{m+1}\arccos \frac {x}{a} + \frac {1}{m+1} \int \frac{x^{m+1}}{\sqrt{a^2-x^2}}dx $ | |
| $ \int x^m \arctan \frac {x}{a}dx=\frac{x^{m+1}}{m+1}\arcsin \frac {x}{a} - \frac {a}{m+1} \int \frac{x^{m+1}}{a^2+x^2}dx $ | |
| $ \int x^m \arccot \frac {x}{a}dx=\frac{x^{m+1}}{m+1}\arcsin \frac {x}{a} + \frac {a}{m+1} \int \frac{x^{m+1}}{a^2+x^2}dx $ | |
| $ \int x^m\arccos \frac {a}{x}dx= \begin{cases} \frac {x^{m+1} \arccos (a/x)}{m+1} - \frac {a}{m+1} \int \frac {x^mdx}{\sqrt{x^2-a^2}} \qquad 0<\arccos \frac{a}{x}< \frac{\pi}{2} \\ \frac {x^{m+1} \arccos (a/x)}{m+1} + \frac {a}{m+1} \int \frac { x^mdx}{\sqrt {x^2-a^2} }\qquad \frac {\pi}{2}<\arccos \frac{a}{x}< \pi \\ \end{cases} $ | |
| $ \int x^m \arcsin \frac {a}{x}dx= \begin{cases} \frac {x^{m+1} \arcsin (a/x)}{m+1} + \frac {a}{m+1} \int \frac {x^m dx}{\sqrt{x^2-a^2}} \qquad 0<\arcsin \frac{a}{x}< \frac{\pi}{2} \\ \frac {x^{m+1} \arcsin (a/x)}{m+1} - \frac {a}{m+1} \int \frac { x^m dx}{\sqrt {x^2-a^2}} \qquad -\frac {\pi}{2}<\arcsin \frac{a}{x}< 0 \\ \end{cases} $ | |
| 27 Integrals Component $ e^{ax} $ | |
| $ \int e^{ax}dx=\frac{e^{ax}}{a} $ | |
| $ \int x e^{ax}dx=\frac{e^{ax}}{a}\left(x-\frac{1}{a} \right) $ | |
| $ \int x^2 e^{ax}dx=\frac{e^{ax}}{a}\left(x^2-\frac{2x}{a}+\frac{2}{a^2}\right) $ | |
| $ \int x^n e^{ax}dx=\frac{x^n e^{ax}}{a}-\frac{n}{a} \int x^{n-1} e^{ax}dx = \frac {e^{ax}}{a} \left( x^n- \frac{nx^{n-1}}{a}+\frac{n(n-1)x^{n-2}}{a^2}- \cdot \cdot \cdot \frac{(-1)^n n!}{a^n} \right ) \qquad \text{if n is a poaitive integer} $ | |
| $ \int \frac {e^{ax}}{x}dx=\ln {x} + \frac {ax}{1 \cdot 1!} + \frac {(ax)^2}{2 \cdot 2!} + \frac {(ax)^3}{3 \cdot 3!} + \cdot \cdot \cdot $ | |
| $ \int \frac {e^{ax}}{x^n}dx = \frac {-e^{ax}}{(n-1)x^{n-1}} + \frac {a}{n-1} \int \frac {e^{ax}}{x^{n-1}}dx $ | |
| $ \int \frac {dx}{p+qe^{ax}}=\frac {x}{p}-\frac {1}{ap} \ln {\left (p+qe^{ax}\right)} $ | |
| $ \int \frac {dx} {\left ( p+qe^{ax} \right) ^2}=\frac {x}{p^2}+\frac {1}{ap(p+qe^{ax})} -\frac{1}{ap^2}\ln {\left (p+qe^{ax}\right)} $ | |
| $ \int \frac {dx}{pe^{ax}+qe^{-ax}}= \begin{cases} \frac {1}{a \sqrt{pq}} \arctan {\left ( \sqrt \frac {p}{q}e^{ax} \right)} \\ \frac {1}{2a \sqrt{-pq}} \ln {\left( \frac{e^{ax}-\sqrt{-q/p}}{e^{ax}+\sqrt{-q/p}} \right)} \\ \end{cases} $ | |
| $ \int e^{ax} \sin bx dx = \frac {e^{ax}(a \sin bx-b \cos bx)}{a^2+b^2} $ | |
| $ \int e^{ax} \cos bx dx = \frac {e^{ax}(a \cos bx-b \sin bx)}{a^2+b^2} $ | |
| $ \int x e^{ax} \sin bx dx = \frac {x e^{ax}(a \sin bx - b \cos bx)}{a^2+b^2} - \frac {e^{ax} \left \{ (a^2-b^2)\sin bx -2ab \cos bx \right \} }{(a^2+b^2)^2} $ | |
| $ \int x e^{ax} \cos bx dx = \frac {x e^{ax}(a \cos bx + b \sin bx)}{a^2+b^2} - \frac {e^{ax} \left \{ (a^2-b^2)\cos bx + 2ab \sin bx \right \} }{(a^2+b^2)^2} $ | |
| $ \int e^{ax} \ln {x} dx = \frac {e^{ax} \ln {x}}{a}-\frac {1}{a} \int \frac {e^{ax}}{x}dx $ | |
| $ \int e^{ax}\sin^n bx dx = \frac{e^{ax} \sin^{n-1}bx}{a^2+n^2 b^2}(a \sin bx -nb \cos bx) + \frac {n(n-1)b^2}{a^2+n^2 b^2} \int e^{ax} \sin^{n-2} bx dx $ | |
| 24 Integrals of lnx | |
| $ \int\ln x dx=x\ln x-x $ | |
| $ \int x\ln x dx=\dfrac{x^{2}}{2}(\ln x-\frac{1}{2}) $ | |
| $ \int x^{m}\ln x dx=\dfrac{x^{m+1}}{m+1}(\ln x-\frac{1}{m+1}) $ | |
| $ \int\dfrac{\ln x}{x} dx=\frac{1}{2}\ln^{2}x $ | |
| $ \int\dfrac{\ln x}{x^{2}} dx=-\dfrac{\ln x}{x}-\dfrac{1}{x} $ | |
| $ \int\ln^{2}x dx=x\ln^{2}x-2x\ln x+2x $ | |
| $ \int\dfrac{\ln^{n}x}{x} dx=\dfrac{\ln^{n+1}x}{n+1} $ | |
| $ \int\dfrac{dx}{x\ln x}=\ln(\ln x) $ | |
| $ \int\dfrac{dx}{\ln x}=\ln(\ln x)+\ln x+\dfrac{\ln^{2}x}{2\cdot2!}+\dfrac{\ln^{3}x}{3\cdot3!}+\cdots $ | |
| $ \int\dfrac{x^{m}dx}{\ln x}=\ln(\ln x)+(m+1)\ln x+\dfrac{(m+1)^{2}\ln^{2}x}{2\cdot2!}+\dfrac{(m+1)^{3}\ln^{3}x}{3\cdot3!}+\cdots $ | |
| $ \int\ln^{n}x dx=x\ln^{n}x-n\int\ln^{n-1}x dx $ | |
| $ \int x^{m}\ln^{n}x dx=\dfrac{x^{m+1}\ln^{n}x}{m+1}-\dfrac{n}{m+1}\int x^{m}\ln^{n-1}x dx $ | |
| $ \int\ln(x^{2}+a^{2}) dx=x\ln(x^{2}+a^{2})-2x+2a\tan^{-1}\dfrac{x}{a} $ | |
| $ \int\ln(x^{2}-a^{2}) dx=x\ln(x^{2}-a^{2})-2x+a\ln(\dfrac{x+a}{x-a}) $ | |
| $ \int x^{m}\ln(x^{2}\pm a^{2}) dx=\dfrac{x^{m}\ln(x^{2}\pm a^{2})}{m+1}-\dfrac{2}{m+1}\int\dfrac{x^{m+2}}{x^{2}\pm a^{2}}dx $ | |
| 25 Integrals of sh ax | |
| $ \int sh ax dx=\dfrac{ch ax}{a} $ | |
| $ \int x sh ax dx=\dfrac{x ch ax}{a}-\dfrac{sh ax}{a^{2}} $ | |
| $ \int x^{2} sh ax dx=(\dfrac{x^{2}}{a^{2}}+\dfrac{2}{a^{3}}) ch ax-\dfrac{2x}{a^{2}} sh ax $ | |
| $ \int\dfrac{sh ax}{x} dx=ax+\dfrac{(ax)^{3}}{3\cdot3!}+\dfrac{(ax)^{5}}{5\cdot5!}+\cdots $ | |
| $ \int\dfrac{sh ax}{x^{2}} dx=- \dfrac{sh ax}{x}+a \int\dfrac{ch ax}{x}dx $ | |
| $ \int\dfrac{dx}{sh ax}=\dfrac{1}{a}\ln th\dfrac{ax}{2} $ | |
| $ \int\dfrac{xdx}{sh ax}=\dfrac{1}{a^{2}}\{ax-\dfrac{(ax)^{3}}{18}+\dfrac{7(ax)^{5}}{1800}-\cdots+\dfrac{2(-1)^{n}(2^{2n}-1)B_{n}(ax)^{2n+1}}{(2n+1)!}\} $ | |
| $ \int sh^{2} ax dx=\dfrac{sh ax ch ax}{2a}-\dfrac{x}{2} $ | |
| $ \int x sh^{2} ax dx=\dfrac{x sh2ax}{4a}-\dfrac{ch2ax}{8a^{2}}-\dfrac{x^{2}}{4} $ | |
| $ \int\dfrac{dx}{sh^{2} ax}=-\dfrac{coth ax}{a} $ | |
| $ \int sh ax sh px dx=\dfrac{sh(a+p) x}{2(a+p)}-\dfrac{sh(a-p)x}{2(a-p)}, p=\pm a $ | |
| $ \int sh ax sin px dx=\dfrac{a ch ax sin px-p sh ax cos px}{a^{2}+p^{2}} $ | |
| $ \int sh ax cos px dx=\dfrac{a ch ax cos px+p sh ax sin px}{a^{2}+p^{2}} $ | |
| $ \int\dfrac{dx}{p+q sh ax}=\dfrac{1}{a\sqrt{p^{2}+q^{2}}}\ln(\dfrac{qe^{ax}+p-\sqrt{p^{2}+q^{2}}}{qe^{ax}+p+\sqrt{p^{2}+q^{2}}}) $ | |
| $ \int\dfrac{dx}{(p+q sh ax)^{2}}=\dfrac{-q ch ax}{a(p^{2}+q^{2})(p+q sh ax)}+\dfrac{p}{p^{2}+q^{2}} \int\dfrac{dx}{p+q sh ax} $ | |
| $ \int\dfrac{dx}{p^{2}+q^{2} sh^{2} ax}=\begin{cases} \dfrac{\dfrac{1}{ap\sqrt{q^{2}-p^{2}}}Arc tg\dfrac{\sqrt{q^{2}-p^{2}} th ax}{p}}{\dfrac{1}{2ap\sqrt{p^{2}-q^{2}}}\ln\biggl(\dfrac{p+\sqrt{p^{2}-q^{2}} th ax}{p-\sqrt{p^{2}-q^{2}} th ax}\biggl)} & .\end{cases}\dfrac{1}{a\sqrt{p^{2}+q^{2}}}\ln\biggl(\dfrac{qe^{ax}+p-\sqrt{p^{2}+q^{2}}}{qe^{ax}+p+\sqrt{p^{2}+q^{2}}}\biggl) $ | |
| $ \int\dfrac{dx}{p^{2}-q^{2} sh^{2} ax}=\dfrac{1}{2ap\sqrt{p^{2}+q^{2}}}\ln(\dfrac{p+\sqrt{p^{2}+q^{2}} th ax}{p-\sqrt{p^{2}+q^{2}} th ax}) $ | |
| $ \int x^{m} sh ax dx=\dfrac{x^{m} ch ax}{a}-\dfrac{m}{a}\int x^{m-1}ch ax dx $ | |
| $ \int sh^{n} ax dx=\dfrac{sh^{n-1} ax ch ax}{an}-\dfrac{n-1}{n}\int sh^{n-2} ax dx $ | |
| $ \int\dfrac{sh ax}{x^{n}} dx=\dfrac{-sh ax}{(n-1)x^{n-1}}+\dfrac{a}{n-1}\int\dfrac{ch ax}{x^{n-1}} dx $ | |
| $ \int\dfrac{dx}{sh^{n} ax}=\dfrac{-ch ax}{a(n-1)sh^{n-1} ax}-\dfrac{n-2}{n-1}{\displaystyle \int}\dfrac{dx}{sh^{n-2} ax} $ | |
| $ \int\dfrac{x}{sh^{n} ax} dx=\dfrac{-x ch ax}{a(n-1)sh^{n-1} ax}-\dfrac{1}{a^{2}(n-1)(n-2) sh^{n-2} ax}-\dfrac{n-2}{n-1}{\displaystyle \int}\dfrac{dx}{sh^{n-2} ax} $ | |
| 26 Integrals of ch ax | |
| $ \int ch ax dx=\dfrac{sh ax}{a} $ | |
| $ \int x ch ax dx=\dfrac{x sh ax}{a}-\dfrac{ch ax}{a^{2}} $ | |
| $ \int x^{2} ch ax dx=(\dfrac{x^{2}}{a^{2}}+\dfrac{2}{a^{3}}) sh ax-\dfrac{2x}{a^{2}} ch ax $ | |
| $ \int\dfrac{ch ax}{x} dx=\ln x+\dfrac{(ax)^{2}}{2\cdot2!}+\dfrac{(ax)^{4}}{4\cdot4!}+\cdots $ | |
| $ \int\dfrac{ch ax}{x^{2}} dx=-\dfrac{ch ax}{x}+a \int\dfrac{sh ax}{x}dx $ | |
| $ \int\dfrac{dx}{ch ax}=\dfrac{2}{a}Arc tg e^{ax} $ | |
| $ \int\dfrac{xdx}{ch ax}=\dfrac{1}{a^{2}}\{\dfrac{(ax)^{2}}{2}-\dfrac{(ax)^{4}}{8}+\dfrac{5(ax)^{6}}{144}-\cdots+\dfrac{(-1)^{n}(2^{2n}-1)E_{n}(ax)^{2n+2}}{(2n+2)!}\} $ | |
| $ \int ch^{2} ax dx=\dfrac{sh ax ch ax}{2a}+\dfrac{x}{2} $ | |
| $ \int x ch^{2} ax dx=\dfrac{x sh2ax}{4a}-\dfrac{ch2ax}{8a^{2}}+\dfrac{x^{2}}{4} $ | |
| $ \int\dfrac{dx}{ch^{2} ax}=\dfrac{th ax}{a} $ | |
| $ \int ch ax ch px dx=\dfrac{sh(a-p) x}{2(a-p)}-\dfrac{sh(a+p)x}{2(a+p)} $ | |
| $ \int ch ax sin px dx=\dfrac{a ch ax sin px-p sh ax cos px}{a^{2}+p^{2}} $ | |
| $ \int ch ax cos px dx=\dfrac{a sh ax cos px+p ch ax sin px}{a^{2}+p^{2}} $ | |
| $ \int\dfrac{dx}{ch ax+1}=\dfrac{1}{a} th\dfrac{ax}{2} $ | |
| $ \int\dfrac{dx}{(ch ax-1)}=-\dfrac{1}{a} coth\dfrac{ax}{2} $ | |
| $ \int\dfrac{xdx}{(ch ax+1)}=\dfrac{x}{a} th\dfrac{ax}{2}-\dfrac{2}{a^{2}}\ln ch\dfrac{ax}{2} $ | |
| $ \int\dfrac{xdx}{(ch ax-1)}=-\dfrac{x}{a}coth\dfrac{ax}{2}+\dfrac{2}{a^{2}}\ln sh\dfrac{ax}{2} $ | |
| $ \int\dfrac{dx}{(ch ax+1)^{2}}=\dfrac{1}{2a}th\dfrac{ax}{2}-\dfrac{1}{6a}th^{3}\dfrac{ax}{2} $ | |
| $ \int\dfrac{dx}{(ch ax-1)^{2}}=\dfrac{1}{2a}coth\dfrac{ax}{2}-\dfrac{1}{6a}coth^{3}\dfrac{ax}{2} $ | |
| $ \int\dfrac{dx}{p+q ch ax}=\begin{cases} \dfrac{\dfrac{2}{a\sqrt{q^{2}-p^{2}}}Arc tg\dfrac{q e^{ax}+p}{\sqrt{q^{2}-p^{2}}}}{\dfrac{1}{a\sqrt{p^{2}-q^{2}}}\ln\biggl(\dfrac{q e^{ax}+p-\sqrt{p^{2}-q^{2}}}{q e^{ax}+p+\sqrt{p^{2}-q^{2}}}\biggl)} & .\end{cases} $ | |
| $ \int\dfrac{dx}{(p+q ch ax)^{2}}=\dfrac{q sh ax}{a(q^{2}-p^{2})(p+q ch ax)}-\dfrac{p}{q^{2}-p^{2}} \int\dfrac{dx}{p+q ch ax} $ | |
| $ \int\dfrac{dx}{p^{2}-q^{2} ch^{2} ax}=\begin{cases} \dfrac{\dfrac{1}{2ap\sqrt{q^{2}-p^{2}}}\ln\biggl(\dfrac{p th ax+\sqrt{p^{2}-q^{2}}}{p th ax-\sqrt{p^{2}-q^{2}}}\biggl)}{\dfrac{1}{ap\sqrt{p^{2}-q^{2}}}-Arc tg\dfrac{p th ax}{\sqrt{q^{2}-p^{2}}}} & .\end{cases} $ | |
| $ \int\dfrac{dx}{p^{2}+q^{2} ch ax}=\begin{cases} \dfrac{\dfrac{1}{2ap\sqrt{p^{2}+q^{2}}}\ln\biggl(\dfrac{p th ax+\sqrt{p^{2}+q^{2}}}{p th ax-\sqrt{p^{2}+q^{2}}}\biggl)}{\dfrac{1}{ap\sqrt{p^{2}+q^{2}}}-Arc tg\dfrac{p th ax}{\sqrt{p^{2}+q^{2}}}} & .\end{cases} $ | |
| $ \int x^{m} ch ax dx=\dfrac{x^{m} sh ax}{a}-\dfrac{m}{a} \int x^{m-1}sh ax dx $ | |
| $ \int ch^{n} ax dx=\dfrac{ch^{n-1} ax sh ax}{an}+\dfrac{n-1}{n} \int ch^{n-2} ax dx $ | |
| $ \int\dfrac{ch ax}{x^{n}} dx=\dfrac{-ch ax}{(n-1)x^{n-1}}+\dfrac{a}{n-1} \int\dfrac{sh ax}{x^{n-1}} dx $ | |
| $ \int\dfrac{dx}{ch^{n} ax}=\dfrac{-sh ax}{a(n-1)ch^{n-1} ax}+\dfrac{n-2}{n-1}{\displaystyle \int}\dfrac{dx}{ch^{n-2} ax} $ | |
| $ \int\dfrac{x}{ch^{n} ax} dx=\dfrac{-x sh ax}{a(n-1)ch^{n-1} ax}+\dfrac{1}{a^{2}(n-1)(n-2) ch^{n-2} ax}+\dfrac{n-2}{n-1}{\displaystyle \int}\dfrac{dx}{ch^{n-2} ax} $ | |
| 27 Integrals of th ax | |
| $ \int th ax dx=\dfrac{\ln ch ax}{a} $ | |
| $ \int th^{2} ax dx=x-\dfrac{th ax}{a} $ | |
| $ \int th^{3} ax dx=\dfrac{1}{a}\dfrac{\ln ch ax}{a}-\dfrac{th^{2} ax}{2a} $ | |
| $ \int\dfrac{th^{n} ax}{ch^{2} ax} dx=\dfrac{th^{n+1} ax}{(n+1)a} $ | |
| $ \int\dfrac{dx}{th ax ch^{2} ax} dx=\dfrac{1}{a}\ln th ax $ | |
| $ \int\dfrac{dx}{th ax} dx=\dfrac{1}{a}\ln sh ax $ | |
| $ \int x th ax dx=\dfrac{1}{a^{2}}\biggl\{\dfrac{(ax)^{3}}{3}-\dfrac{(ax)^{5}}{15}+\dfrac{2(ax)^{7}}{105}\cdots+\dfrac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_{n}(ax)^{2n+1}}{(2n+1)|}\biggl\} $ | |
| $ \int x th^{2} ax dx=\dfrac{x^{2}}{2}-\dfrac{x th ax}{a}+\dfrac{1}{a^{2}}\ln ch ax $ | |
| $ \int\dfrac{th ax}{x} dx=\biggl\{ ax-\dfrac{(ax)^{3}}{9}+\dfrac{2(ax)^{5}}{75}-\cdots+\dfrac{(-1)^{n-1}2^{2n}(2^{2n}-1)B_{n}(ax)^{2n-1}}{(2n-1)(2n)!}\biggl\} $ | |
| $ \int\dfrac{dx}{p+q th ax}=\dfrac{px}{p^{2}-q^{2}}-\dfrac{q}{a(p^{2}-q^{2})}\ln(q sh ax+p ch ax) $ | |
| $ \int th^{n} ax dx=-\dfrac{th^{n+1} ax}{a(n-1)}+ \int th^{n-2} ax dx $ | |
| 28 Integrals of coth ax | |
| $ \int coth ax dx=\dfrac{\ln sh ax}{a} $ | |
| $ \int coth^{2} ax dx=x-\dfrac{coth ax}{a} $ | |
| $ \int coth^{3} ax dx=\dfrac{1}{a}\dfrac{\ln sh ax}{a}-\dfrac{coth^{2} ax}{2a} $ | |
| $ \int\dfrac{coth^{n} ax}{sh^{2} ax} dx=\dfrac{coth^{n+1} ax}{(n+1)a} $ | |
| $ \int\dfrac{dx}{coth ax sh^{2} ax} dx=\dfrac{1}{a}\ln coth ax $ | |
| $ \int\dfrac{dx}{coth ax} dx=\dfrac{1}{a}\ln ch ax $ | |
| $ \int x coth ax dx=\dfrac{1}{a^{2}}\biggl\{ ax+\dfrac{(ax)^{3}}{9}-\dfrac{(ax)^{5}}{225}+\dfrac{2(ax)^{7}}{105}+\cdots\dfrac{(-1)^{n-1}2^{2n}B_{n}(ax)^{2n+1}}{(2n+1)|}\biggl\} $ | |
| $ \int x coth^{2} ax dx=\dfrac{x^{2}}{2}-\dfrac{x coth ax}{a}+\dfrac{1}{a^{2}}\ln sh ax $ | |
| $ \int\dfrac{coth ax}{x} dx=\biggl\{-\dfrac{1}{ax}+\dfrac{ax}{3}-\dfrac{(ax)^{3}}{135}+\cdots\dfrac{(-1)^{n-1}2^{2n}B_{n}(ax)^{2n-1}}{(2n-1)(2n)!}\biggl\} $ | |
| $ \int\dfrac{dx}{p+q coth ax}=\dfrac{px}{p^{2}-q^{2}}-\dfrac{q}{a(p^{2}-q^{2})}\ln(p sh ax+q ch ax) $ | |
| $ \int coth^{n} ax dx=-\dfrac{coth^{n-1} ax}{a(n-1)}+ \int coth^{n-2} ax dx $ | |
| 29 Integrals of $ \dfrac{1}{ch ax} $ | |
| $ \int\dfrac{1}{ch ax}dx=\dfrac{2}{a}Arc tg e^{ax} $ | |
| $ \int\dfrac{1}{ch^{2} ax}dx=\dfrac{th ax}{a} $ | |
| $ \int\dfrac{1}{ch^{3} ax}dx=\dfrac{th ax}{2a ch ax}+\dfrac{1}{2a}Arc tg sh ax $ | |
| $ \int\dfrac{th ax}{ch^{n} ax}dx=-\dfrac{1}{na ch^{n} ax} $ | |
| $ \int ch ax dx=\dfrac{sh ax}{a} $ | |
| $ \int\dfrac{xdx}{ch ax}=\dfrac{1}{a^{2}}\biggl\{\dfrac{(ax)^{3}}{2}-\dfrac{(ax)^{4}}{8}+\dfrac{5(ax)^{6}}{144}+\cdots+\dfrac{(-1)^{n}E_{n}(ax)^{2n+2}}{(2n+2)(2n)|}\biggl\} $ | |
| $ \int\dfrac{xdx}{ch^{2} ax}=\dfrac{x th ax}{a}-\dfrac{1}{a^{2}}\ln ch ax $ | |
| $ \int\dfrac{dx}{x sh ax}=\ln x-\dfrac{(ax)^{2}}{4}+\dfrac{5(ax)^{4}}{96}-\dfrac{6(ax)^{6}}{4320}+\cdots\dfrac{(-1)^{n}E_{n}(ax)^{2n}}{2n(2n)|}\biggl\} $ | |
| $ \int\dfrac{dx}{q+\dfrac{p}{ch ax}}=\dfrac{x}{q}-\dfrac{p}{q} \int\dfrac{dx}{p+q ch ax} $ | |
| $ \int\dfrac{1}{ch^{n} ax}dx=\dfrac{th ax}{a(n-1) ch^{n-2} ax}+\dfrac{(n-2)}{(n-1)} \int\dfrac{dx}{ch^{n-2} ax} $ | |
| 30 Integrals of 1/sh ax | |
| $ \int\dfrac{1}{sh ax}dx=\dfrac{1}{a}\ln th\dfrac{ax}{2} $ | |
| $ \int\dfrac{1}{sh^{2} ax}dx=-\dfrac{coth ax}{a} $ | |
| $ \int\dfrac{1}{sh^{3} ax}dx=\dfrac{coth ax}{2a sh ax}+\dfrac{1}{2a}\ln th\dfrac{ax}{2} $ | |
| $ \int\dfrac{coth ax}{sh^{n} ax}dx=-\dfrac{1}{na sh^{n} ax} $ | |
| $ \int sh ax dx=\dfrac{ch ax}{a} $ | |
| $ \int\dfrac{xdx}{sh ax}=\dfrac{1}{a^{2}}\biggl\{ ax-\dfrac{(ax)^{3}}{18}+\dfrac{7(ax)^{5}}{1800}+\cdots+\dfrac{2(-1)^{n}(2^{2n-1}-1)B_{n}(ax)^{2n+1}}{(2n+1)|}\biggl\} $ | |
| $ \int\dfrac{xdx}{sh^{2} ax}=\dfrac{x coth ax}{a}-\dfrac{1}{a^{2}}\ln sh ax $ | |
| $ \int\dfrac{dx}{x sh ax}=-\dfrac{1}{ax}-\dfrac{ax}{6}+\dfrac{7(ax)^{3}}{1080}+\cdots\dfrac{(-1)^{n}(2^{2n-1}-1)B_{n}(ax)^{2n-1}}{(2n-1)(2n)|}\biggl\} $ | |
| $ \int\dfrac{dx}{q+\dfrac{p}{sh ax}}=\dfrac{x}{q}-\dfrac{p}{q} \int\dfrac{dx}{p+q sh ax} $ | |
| $ \int\dfrac{1}{sh^{n} ax}dx=\dfrac{coth ax}{a(n-1) sh^{n-2} ax}-\dfrac{(n-2)}{(n-1)} \int\dfrac{dx}{sh^{n-2} ax} $ | |
| 31 Integrals of sh ax et ch ax | |
| $ \int sh ax ch ax dx=\dfrac{sh^{2} ax}{2a} $ | |
| $ \int sh px ch qx dx=\dfrac{ch(p+q)x}{2(p+q)}+\dfrac{ch(p-q)x}{2(p-q)} $ | |
| $ \int sh^{n} ax ch ax dx=\dfrac{sh^{n+1} ax}{(n+1)a} $ | |
| $ \int ch^{n} ax sh ax dx=\dfrac{ch^{n+1} ax}{(n+1)a} $ | |
| $ \int sh^{2} ax ch^{2} ax dx=\dfrac{sh4ax}{32a}-\dfrac{x}{8} $ | |
| $ \int\dfrac{dx}{sh ax ch ax}=\dfrac{1}{a}\ln th ax $ | |
| $ \int\dfrac{dx}{sh^{2} ax ch ax}=-\dfrac{1}{a}Arc tg sh ax-\dfrac{1}{a sh ax} $ | |
| $ \int\dfrac{dx}{sh ax ch^{2} ax}=\dfrac{1}{a ch ax}+\dfrac{1}{a}\ln th\dfrac{ax}{2} $ | |
| $ \int\dfrac{dx}{sh^{2} ax ch^{2} ax}=-\dfrac{2 coth2ax}{a} $ | |
| $ \int\dfrac{sh^{2} ax dx}{ch ax}=-\dfrac{1}{a}Arc tg sh ax+\dfrac{sh ax}{a} $ | |
| $ \int\dfrac{ch^{2} ax dx}{sh ax}=\dfrac{1}{a}\ln th\dfrac{ax}{2}+\dfrac{ch ax}{a} $ | |
| $ \int\dfrac{dx}{sh ax(ch ax+1)}=\dfrac{1}{2a}\ln th\dfrac{ax}{2}+\dfrac{1}{2a(ch ax+1)} $ | |
| $ \int\dfrac{dx}{(sh ax+1) ch ax}=\dfrac{1}{2a}\ln\biggl(\dfrac{1+sh ax}{ch ax}\biggl)+\dfrac{1}{a}Arc tg e^{ax} $ | |
| $ \int\dfrac{dx}{sh ax(ch ax-1)}=-\dfrac{1}{2a}\ln th\dfrac{ax}{2}-\dfrac{1}{2a(ch ax-1)} $ | |
| 32 Integrals of Hyperbolic Inverse functions : arg sh ax | |
| $ \int\arg sh\dfrac{x}{a}dx=x\arg sh\dfrac{x}{a}-\sqrt{x^{2}+a^{2}} $ | |
| $ \int x\arg sh\dfrac{x}{a} dx=\biggl(\dfrac{x^{2}}{2}+\dfrac{a^{2}}{4}\biggl)\arg sh\dfrac{x}{a}-\dfrac{x\sqrt{x^{2}+a^{2}}}{4} $ | |
| $ \int x^{2}\arg sh\dfrac{x}{a} dx=\dfrac{x^{3}}{3}\arg sh\dfrac{x}{a}+\dfrac{(2a^{2-}x^{2})\sqrt{x^{2}+a^{2}}}{9} $ | |
| $ \int\dfrac{\arg sh\dfrac{x}{a}}{x}dx=\Biggl\{\begin{array}{c} \dfrac{x}{a}-\dfrac{(\dfrac{x}{a})^{3}}{2\cdot3\cdot3}+\dfrac{1\cdot3(\dfrac{x}{a})^{5}}{2\cdot4\cdot5\cdot5}-\dfrac{1\cdot3\cdot5(\dfrac{x}{a})^{7}}{2\cdot4\cdot6\cdot7\cdot7}+\cdots,|x|<a\\ \dfrac{\ln^{2}(\dfrac{2x}{a})}{2}-\dfrac{(\dfrac{a}{x})^{2}}{2\cdot2\cdot2}+\dfrac{1\cdot3(\dfrac{a}{x})^{4}}{2\cdot4\cdot4\cdot4}-\dfrac{1\cdot3\cdot5(\dfrac{a}{x})^{6}}{2\cdot4\cdot6\cdot6\cdot6}+\cdots, x>a\\ \dfrac{\ln^{2}(\dfrac{-2x}{a})}{2}+\dfrac{(\dfrac{a}{x})^{2}}{2\cdot2\cdot2}-\dfrac{1\cdot3(\dfrac{a}{x})^{4}}{2\cdot4\cdot4\cdot4}+\dfrac{1\cdot3\cdot5(\dfrac{a}{x})^{6}}{2\cdot4\cdot6\cdot6\cdot6}+\cdots, x<-a\end{array} $ | |
| $ \int\dfrac{\arg sh\dfrac{x}{a}}{x^{2}}dx=-\dfrac{\arg sh\dfrac{x}{a}}{x}-\dfrac{1}{a}\ln\Biggl(\dfrac{a+\sqrt{x^{2}+a^{2}}}{x}\Biggl) $ | |
| 33 Integrals of Hyperbolic Inverse functions : arg ch ax | |
| $ \int\arg ch\dfrac{x}{a}dx=\begin{cases} \dfrac{x\arg ch\dfrac{x}{a}-\sqrt{x^{2}-a^{2}}}{x\arg sh\dfrac{x}{a}+\sqrt{x^{2}-a^{2}}} & .\end{cases} $ | |
| $ \int x\arg ch\dfrac{x}{a} dx=\begin{cases} \dfrac{\frac{1}{4}(2x^{2}-a^{2})\arg ch\dfrac{x}{a}-\frac{1}{4}x\sqrt{x^{2}-a^{2}}}{\frac{1}{4}(2x^{2}-a^{2})\arg ch\dfrac{x}{a}+\frac{1}{4}x\sqrt{x^{2}-a^{2}}} & .\end{cases} $ | |
| $ \int x^{2}\arg ch\dfrac{x}{a} dx=\begin{cases} \dfrac{\frac{1}{3}x^{3}\arg ch\dfrac{x}{a}-\frac{1}{9}(x^{2}+2a^{2})\sqrt{x^{2}-a^{2}}}{\frac{1}{3}x^{3}\arg ch\dfrac{x}{a}+\frac{1}{9}(x^{2}+2a^{2})\sqrt{x^{2}-a^{2}}} & .\end{cases} $ | |
| $ \int\dfrac{\arg ch\dfrac{x}{a}}{x}dx=\pm\Biggl[\dfrac{\ln^{2}(\dfrac{2x}{a})}{2}+\dfrac{(\dfrac{a}{x})^{2}}{2\cdot2\cdot2}+\dfrac{1\cdot3(\dfrac{a}{x})^{4}}{2\cdot4\cdot4\cdot4}+\dfrac{1\cdot3\cdot5(\dfrac{a}{x})^{6}}{2\cdot4\cdot6\cdot6\cdot6}+\cdots,\Biggl] $ | |
| $ \int\dfrac{\arg ch\dfrac{x}{a}}{x^{2}}dx=-\dfrac{\arg ch\dfrac{x}{a}}{x}\mp\dfrac{1}{a}\ln\Biggl(\dfrac{a+\sqrt{x^{2}+a^{2}}}{x}\Biggl) $ | |
| 34 Integrals of Hyperbolic Inverse functions : arg th ax | |
| $ \int\arg th\dfrac{x}{a}dx=x\arg th\dfrac{x}{a}+\dfrac{a}{2}\ln(a^{2}-x^{2}) $ | |
| $ \int x\arg th\dfrac{x}{a} dx=\dfrac{ax}{2}+\frac{1}{2}(x^{2}-a^{2})\arg th\dfrac{x}{a} $ | |
| $ \int x^{2}\arg th\dfrac{x}{a} dx=\dfrac{ax^{2}}{6}+\frac{a^{3}}{6}\ln(a^{2}-x^{2})+\dfrac{x^{3}}{3}\arg th\dfrac{x}{a} $ | |
| $ \int\dfrac{\arg th\dfrac{x}{a}}{x}dx=\dfrac{x}{a}+\dfrac{(\dfrac{x}{a})^{3}}{3^{2}}+\dfrac{(\dfrac{x}{a})^{5}}{5^{2}}+\cdots $ | |
| $ \int\dfrac{\arg th\dfrac{x}{a}}{x^{2}}dx=-\dfrac{\arg th\dfrac{x}{a}}{x}+\dfrac{1}{2a}\ln\Biggl(\dfrac{x^{2}}{a^{2}-x^{2}}\Biggl) $ | |
| 35 Integrals of Hyperbolic Inverse functions : arg coth ax | |
| $ \int\arg coth\dfrac{x}{a}dx=x\arg coth x+\dfrac{a}{2}\ln(x^{2}-a^{2}) $ | |
| $ \int x\arg coth\dfrac{x}{a} dx=\dfrac{ax}{2}+\frac{1}{2}(x^{2}-a^{2})\arg coth\dfrac{x}{a} $ | |
| $ \int x^{2}\arg coth\dfrac{x}{a} dx=\dfrac{ax^{2}}{6}+\frac{a^{3}}{6}\ln(x^{2}-a^{2})+\dfrac{x^{3}}{3}\arg coth\dfrac{x}{a} $ | |
| $ \int\dfrac{\arg coth\dfrac{x}{a}}{x}dx=-\Biggl(\dfrac{a}{x}+\dfrac{(\dfrac{a}{x})^{3}}{3^{2}}+\dfrac{(\dfrac{a}{x})^{5}}{5^{2}}+\cdots\Biggl) $ | |
| $ \int\dfrac{\arg coth\dfrac{x}{a}}{x^{2}}dx=-\dfrac{\arg coth\dfrac{x}{a}}{x}+\dfrac{1}{2a}\ln\Biggl(\dfrac{x^{2}}{x^{2}-a^{2}}\Biggl) $ | |
| 36 Integrals of a/x | |
| $ \int\arg ch\dfrac{a}{x}dx=\begin{cases} \dfrac{x\arg ch\dfrac{a}{x}+\arcsin\dfrac{x}{a}}{x\arg ch\dfrac{a}{x}-\arcsin\dfrac{x}{a}} & .\end{cases} $ | |
| $ \int x\arg ch\dfrac{a}{x} dx=\begin{cases} \dfrac{\frac{1}{2}x^{2}\arg ch\dfrac{a}{x}-\dfrac{1}{2}a\sqrt{a^{2}-x^{2}}}{\frac{1}{2}x^{2}\arg ch\dfrac{a}{x}+\dfrac{1}{2}a\sqrt{a^{2}-x^{2}}} & .\end{cases} $ | |
| $ \int\dfrac{\arg ch\dfrac{a}{x}}{x}dx=\begin{cases} \dfrac{\dfrac{-\frac{1}{2}\ln(\dfrac{a}{x})\ln(\dfrac{4a}{x})}{2}-\dfrac{(\dfrac{x}{a})^{2}}{2\cdot2\cdot2}-\dfrac{1\cdot3(\dfrac{x}{a})^{4}}{2\cdot4\cdot4\cdot4}-\cdots}{\dfrac{\frac{1}{2}\ln(\dfrac{a}{x})\ln(\dfrac{4a}{x})}{2}+\dfrac{(\dfrac{x}{a})^{2}}{2\cdot2\cdot2}+\dfrac{1\cdot3(\dfrac{x}{a})^{4}}{2\cdot4\cdot4\cdot4}+\cdots} & .\end{cases} $ | |
| $ \int\arg sh\dfrac{a}{x}dx=x\arg sh\dfrac{a}{x}\pm\arg sh\dfrac{x}{a} $ | |
| $ \int x\arg sh\dfrac{a}{x} dx=\dfrac{x^{2}}{2}\arg sh\dfrac{a}{x}\pm\dfrac{1}{2}a\sqrt{a^{2}+x^{2}} $ | |
| $ \int\dfrac{\arg sh\dfrac{a}{x}}{x}dx=\Biggl\{\begin{array}{c} \dfrac{\frac{1}{2}\ln(\dfrac{x}{a})\ln(\dfrac{4a}{x})}{2}+\dfrac{(\dfrac{x}{a})^{2}}{2\cdot2\cdot2}-\dfrac{1\cdot3(\dfrac{x}{a})^{4}}{2\cdot4\cdot4\cdot4}+\cdots\\ \dfrac{\frac{1}{2}\ln(\dfrac{x}{a})\ln(\dfrac{4a}{x})}{2}-\dfrac{(\dfrac{x}{a})^{2}}{2\cdot2\cdot2}+\dfrac{1\cdot3(\dfrac{x}{a})^{4}}{2\cdot4\cdot4\cdot4}+\cdots\\ -\frac{a}{x}+\dfrac{(\dfrac{a}{x})^{3}}{2\cdot3\cdot3}-\dfrac{1\cdot3(\dfrac{a}{x})^{5}}{2\cdot4\cdot5\cdot5}+\cdots\end{array} $ | |
| $ \int x^{m}\arg sh\dfrac{x}{a} dx=\dfrac{x^{m+1}}{m+1}\arg sh\dfrac{x}{a}-\dfrac{1}{m+1} \int\dfrac{x^{m+1}}{\sqrt{x^{2}+a^{2}}}dx $ | |
| $ \int x^{m}\arg ch\dfrac{x}{a} dx=\begin{cases} \dfrac{\dfrac{x^{m+1}}{m+1}\arg ch\dfrac{x}{a}-\dfrac{1}{m+1} \int\dfrac{x^{m+1}}{\sqrt{x^{2}-a^{2}}}dx}{\dfrac{x^{m+1}}{m+1}\arg ch\dfrac{x}{a}+\dfrac{1}{m+1} \int\dfrac{x^{m+1}}{\sqrt{x^{2}-a^{2}}}dx} & .\end{cases} $ | |
| $ \int x^{m}\arg th\dfrac{x}{a} dx=\dfrac{x^{m+1}}{m+1}\arg th\dfrac{x}{a}-\dfrac{a}{m+1} \int\dfrac{x^{m+1}}{a^{2}-x^{2}}dx $ | |
| $ \int x^{m}\arg coth\dfrac{x}{a} dx=\dfrac{x^{m+1}}{m+1}\arg coth\dfrac{x}{a}-\dfrac{a}{m+1} \int\dfrac{x^{m+1}}{a^{2}-x^{2}}dx $ | |
| $ \int x^{m}\arg ch\dfrac{a}{x} dx=\begin{cases} \dfrac{\dfrac{x^{m+1}}{m+1}\arg ch\dfrac{a}{x}+\dfrac{a}{m+1} \int\dfrac{x^{m}}{\sqrt{a^{2}-x^{2}}}dx}{\dfrac{x^{m+1}}{m+1}\arg ch\dfrac{a}{x}-\dfrac{a}{m+1} \int\dfrac{x^{m}}{\sqrt{a^{2}-x^{2}}}dx} & .\end{cases} $ | |
| $ \int x^{m}\arg sh\dfrac{a}{x} dx=\dfrac{x^{m+1}}{m+1}\arg sh\dfrac{x}{a}\pm\dfrac{a}{m+1} \int\dfrac{x^{m}}{\sqrt{x^{2}+a^{2}}}dx $ | |
| Particular Integral, componant $ x^{2}-a^{2},x^{2}<a^{2} $ | |
| $ \int \dfrac{dx}{a^{2}-x^{2}} = \dfrac{1}{2a}\ln\left(\dfrac{a+x}{a-x}\right) \qquad o\grave{u}\qquad \dfrac{1}{a} Arg tg \dfrac{x}{a} $ | |
| $ \int \dfrac{xdx}{a^2 - x^2} = -\dfrac{1}{2} \ln\left({a^2-x^2}\right) $ | |
| $ \int \dfrac{x^2 dx}{a^2 - x^2} = -x + \dfrac{a}{2}\ln\left(\dfrac{a+x}{a-x}\right) $ | |
| $ \int \dfrac{x^3 dx}{a^2 - x^2} = -\dfrac{x^2}{2} - \dfrac{a^2}{2}\ln\left({a^2-x^2}\right) $ | |
| $ \int \dfrac{dx}{x\left(a^2-x^2\right)} = \dfrac{1}{2a^2}\ln\left(\dfrac{x^2}{a^2-x^2}\right) $ | |
| $ \int \dfrac{dx}{x^2\left(a^2-x^2\right)} = -\dfrac{1}{a^2x} + \dfrac{1}{2a^3}\ln\left(\dfrac{a+x}{a-x}\right) $ | |
| $ \int \dfrac{dx}{x^3\left(a^2-x^2\right)} = -\dfrac{1}{2a^2x^2} + \dfrac{1}{2a^4}\ln\left(\dfrac{x^2}{a^2-x^2}\right) $ | |
| $ \int \dfrac{dx}{\left(a^2-x^2\right)^2} = \dfrac{x}{2a^2\left(a^2-x^2\right)} + \dfrac{1}{4a^3}\ln\left(\dfrac{a+x}{a-x}\right) $ | |
| $ \int \dfrac{xdx}{\left(a^2-x^2\right)^2} = \dfrac{1}{2\left(a^2-x^2\right)} $ | |
| $ \int \dfrac{x^2dx}{\left(a^2-x^2\right)^2} = \dfrac{x}{2\left(a^2-x^2\right)} - \dfrac{1}{4a}\ln\left(\dfrac{a+x}{a-x}\right) $ | |
| $ \int \dfrac{x^3dx}{\left(a^2-x^2\right)^2} = \dfrac{a^2}{2\left(a^2-x^2\right)} - \dfrac{1}{2}\ln\left(a^2-x^2\right) $ | |
| $ \int \dfrac{dx}{x\left(a^2-x^2\right)^2} = \dfrac{1}{2a^2\left(a^2-x^2\right)} - \dfrac{1}{2a^4}\ln\left(\dfrac{x^2}{a^2-x^2}\right) $ | |
| $ \int \dfrac{dx}{x^2\left(a^2-x^2\right)^2} = \dfrac{-1}{a^4x} + \dfrac{x}{2a^4\left(a^2-x^2\right)} - \dfrac{3}{4a^5}\ln\left(\dfrac{a+x}{a-x}\right) $ | |
| $ \int \dfrac{dx}{x^3\left(a^2-x^2\right)^2} = \dfrac{-1}{2a^4x^2} + \dfrac{1}{2a^4\left(a^2-x^2\right)} + \dfrac{1}{a^6}\ln\left(\dfrac{x^2}{a^2-x^2}\right) $ | |
| $ \int \dfrac{dx}{\left(a^2-x^2\right)^n} = \dfrac{x}{2\left(n-1\right)a^2\left(a^2-x^2\right)^{n-1}} + \dfrac{2n-3}{\left(2n-2\right)a^2} \int\dfrac{dx}{\left(a^2-x^2\right)^{n-1}} $ | |
| $ \int \dfrac{xdx}{\left(a^2-x^2\right)^n} = \dfrac{1}{2\left(n-1\right)\left(a^2-x^2\right)^{n-1}} $ | |
| $ \int \dfrac{dx}{x\left(a^2-x^2\right)^n} = \dfrac{1}{2\left(n-1\right)\left(a^2-x^2\right)^{n-1}} + \dfrac{1}{a^2}\int\dfrac{dx}{x\left(a^2-x^2\right)^{n-1}} $ | |
| $ \int \dfrac{x^mdx}{\left(a^2-x^2\right)^n} = a^2\int\dfrac{x^{m-2}dx}{\left(a^2-x^2\right)^{n-1}} - \int\dfrac{x^{m-2}dx}{\left(a^2-x^2\right)^{n-1}} $ | |
| $ \int \dfrac{dx}{x^m\left(a^2-x^2\right)^n} = \dfrac{1}{a^2}\int\dfrac{dx}{x^m\left(a^2-x^2\right)^{n-1}} + \dfrac{1}{a^2}\int\dfrac{dx}{x^{m-2}\left(a^2-x^2\right)^n} $ | |
| Particular Integral, componant $ \sqrt{x^2+a^2} $ | |
| $ \int \dfrac{dx}{\sqrt{x^2+a^2}} = \ln\left(x+\sqrt{x^2+a^2}\right) \qquad o\grave{u}\qquad Arg sh \dfrac{x}{a} $ | |
| $ \int \dfrac{xdx}{\sqrt{x^2+a^2}} = \sqrt{x^2+a^2} $ | |
| $ \int \dfrac{x^2dx}{\sqrt{x^2+a^2}} = \dfrac{x\sqrt{x^2+a^2}}{2} - \dfrac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right) $ | |
| $ \int \dfrac{x^3dx}{\sqrt{x^2+a^2}} = \dfrac{\left(x^2+a^2\right)^{3/2}}{3} - a^2\sqrt{x^2+a^2} $ | |
| $ \int \dfrac{dx}{x\sqrt{x^2+a^2}} = -\dfrac{1}{a} \ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right) $ | |
| $ \int \dfrac{dx}{x^2\sqrt{x^2+a^2}} = -\dfrac{\sqrt{x^2+a^2}}{a^2x} $ | |
| $ \int \dfrac{dx}{x^3\sqrt{x^2+a^2}} = -\dfrac{\sqrt{x^2+a^2}}{2a^2x^2} + \dfrac{1}{2a^3}\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right) $ | |
| $ \int \sqrt{x^2+a^2}dx = \dfrac{x\sqrt{x^2+a^2}}{2} + \dfrac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right) $ | |
| $ \int x\sqrt{x^2+a^2}dx = \dfrac{\left(x^2+a^2\right)^{3/2}}{3} $ | |
| $ \int x^2\sqrt{x^2+a^2}dx = \dfrac{x\left(x^2+a^2\right)^{3/2}}{4} - \dfrac{a^2x\sqrt{x^2+a^2}}{8} - \dfrac{a^4}{8}\ln\left(x+\sqrt{x^2+a^2}\right) $ | |
| $ \int x^3\sqrt{x^2+a^2}dx = \dfrac{\left(x^2+a^2\right)^{5/2}}{5} - \dfrac{a^2\left(x^2+a^2\right)^{3/2}}{3} $ | |
| $ \int \dfrac{\sqrt{x^2+a^2}}{x}dx = \sqrt{x^2+a^2} - a\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right) $ | |
| $ \int \dfrac{\sqrt{x^2+a^2}}{x^2}dx = -\dfrac{\sqrt{x^2+a^2}}{x} + \ln\left(x+\sqrt{x^2+a^2}\right) $ | |
| $ \int \dfrac{\sqrt{x^2+a^2}}{x^3}dx = -\dfrac{\sqrt{x^2+a^2}}{2x^2} - \dfrac{1}{2a}\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right) $ | |
| $ \int \dfrac{dx}{\left(x^2+a^2\right)^{3/2}} = \dfrac{x}{a^2\sqrt{x^2+a^2}} $ | |
| $ \int \dfrac{xdx}{\left(x^2+a^2\right)^{3/2}} = \dfrac{-1}{\sqrt{x^2+a^2}} $ | |
| $ \int \dfrac{x^2dx}{\left(x^2+a^2\right)^{3/2}} = \dfrac{-x}{\sqrt{x^2+a^2}} +\ln\left(x+\sqrt{x^2+a^2}\right) $ | |
| $ \int \dfrac{x^3dx}{\left(x^2+a^2\right)^{3/2}} = \sqrt{x^2+a^2} + \dfrac{a^2}{\sqrt{x^2+a^2}} $ | |
| $ \int \dfrac{dx}{x\left(x^2+a^2\right)^{3/2}} = \dfrac{1}{a^2\sqrt{x^2+a^2}} - \dfrac{1}{a^3}\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right) $ | |
| $ \int \dfrac{dx}{x^2\left(x^2+a^2\right)^{3/2}} = -\dfrac{\sqrt{x^2+a^2}}{a^4x} - \dfrac{x}{a^4\sqrt{x^2+a^2}}{x} $ | |
| $ \int \dfrac{dx}{x^3\left(x^2+a^2\right)^{3/2}} = \dfrac{-1}{2a^2x^2\sqrt{x^2+a^2}} - \dfrac{3}{2a^4\sqrt{x^2+a^2}} + \dfrac{3}{2a^5}\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right) $ | |
| $ \int \left(x^2+a^2\right)^{3/2}dx = \dfrac{x\left(x^2+a^2\right)^{3/2}}{4} + \dfrac{3a^2x\sqrt{x^2+a^2}}{8} + \dfrac{3}{8}a^4\ln\left(x+\sqrt{x^2+a^2}\right) $ | |
| $ \int x\left(x^2+a^2\right)^{3/2}dx = \dfrac{\left(x^2+a^2\right)^{5/2}}{5} $ | |
| $ \int x^2\left(x^2+a^2\right)^{3/2}dx = \dfrac{x\left(x^2+a^2\right)^{5/2}}{6} - \dfrac{a^2x\left(x^2+a^2\right)^{3/2}}{24} - \dfrac{a^4x\sqrt{x^2+a^2}}{16} + \dfrac{a^6}{16}\ln\left(x+\sqrt{x^2+a^2}\right) $ | |
| $ \int x^3\left(x^2+a^2\right)^{3/2}dx = \dfrac{\left(x^2+a^2\right)^{7/2}}{7} - \dfrac{a^2\left(x^2+a^2\right)^{5/2}}{5} $ | |
| $ \int \dfrac{\left(x^2+a^2\right)^{3/2}}{x}dx = \dfrac{\left(x^2+a^2\right)^{3/2}}{3} + a^2\sqrt{x^2+a^2} - a^3\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right) $ | |
| $ \int \dfrac{\left(x^2+a^2\right)^{3/2}}{x^2}dx = - \dfrac{\left(x^2+a^2\right)^{3/2}}{x} + \dfrac{3x\sqrt{x^2+a^2}}{2} + \dfrac{3}{2}a^2\ln\left(x+\sqrt{x^2+a^2}\right) $ | |
| $ \int \dfrac{\left(x^2+a^2\right)^{3/2}}{x^3}dx = - \dfrac{\left(x^2+a^2\right)^{3/2}}{2x^2} + \dfrac{3}{2}\sqrt{x^2+a^2} - \dfrac{3}{2}a\ln\left(\dfrac{a+\sqrt{x^2+a^2}}{x}\right) $ | |
| Particular Integral, componant $ \sqrt{x^2-a^2} $ | |
| $ \int \dfrac{dx}{\sqrt{x^2-a^2}} = \ln\left(x+\sqrt{x^2-a^2}\right) \qquad o\grave{u} \qquad argcosh \dfrac{x}{a}, \qquad \int \dfrac{xdx}{\sqrt{x^2-a^2}} = \sqrt{x^2-a^2} $ | |
| $ \int \dfrac{x^2dx}{\sqrt{x^2-a^2}} = \dfrac{x\sqrt{x^2-a^2}}{2} + \dfrac{a^2}{2} \ln\left(x+\sqrt{x^2-a^2}\right) $ | |
| $ \int \dfrac{x^3dx}{\sqrt{x^2-a^2}} = \dfrac{\left(x^2-a^2\right)^{3/2}}{3} + a^2\sqrt{x^2-a^2} $ | |
| $ \int \dfrac{dx}{x\sqrt{x^2-a^2}} = \dfrac{1}{a} \arccos \left|\frac{a}{x}\right \vert $ | |
| $ \int \dfrac{dx}{x^2\sqrt{x^2-a^2}} = \dfrac{\sqrt{x^2-a^2}}{a^2x} $ | |
| $ \int \dfrac{dx}{x^3\sqrt{x^2-a^2}} =\dfrac{\sqrt{x^2-a^2}}{2a^2x^2} + \dfrac{1}{2a^3} \arccos \left|\frac{a}{x}\right \vert $ | |
| $ \int \sqrt{x^2-a^2} dx = \dfrac{x\sqrt{x^2-a^2}}{2} - \dfrac{a^2}{2} \ln\left(x+\sqrt{x^2-a^2}\right) $ | |
| $ \int x\sqrt{x^2-a^2}dx = \dfrac{\left(x^2-a^2\right)^{3/2}}{3} $ | |
| $ \int x^2\sqrt{x^2-a^2}dx = \dfrac{x\left(x^2-a^2\right)^{3/2}}{4} + \dfrac{a^2 x\sqrt{x^2-a^2}}{8} - \dfrac{a^4}{8} \ln\left(x+\sqrt{x^2-a^2}\right) $ | |
| $ \int x^3\sqrt{x^2-a^2}dx = \dfrac{\left(x^2-a^2\right)^{5/2}}{5} + \dfrac{a^2\left(x^2-a^2\right)^{3/2}}{3} $ | |
| $ \int \dfrac{\sqrt{x^2-a^2}}{x} dx =\sqrt{x^2-a^2} - a \arccos \left|\frac{a}{x}\right \vert $ | |
| $ \int \dfrac{\sqrt{x^2-a^2}}{x^2} dx = - \dfrac{\sqrt{x^2-a^2}}{x} + \ln\left(x+\sqrt{x^2-a^2}\right) $ | |
| $ \int \dfrac{\sqrt{x^2-a^2}}{x^3} dx = - \dfrac{\sqrt{x^2-a^2}}{2x^2} + \dfrac{1}{2a} \arccos \left|\frac{a}{x}\right \vert $ | |
| $ \int \dfrac{dx}{\left(x^2-a^2\right)^{3/2}} = \dfrac{x}{a^2\sqrt{x^2-a^2}} $ | |
| $ \int \dfrac{xdx}{\left(x^2-a^2\right)^{3/2}} = \dfrac{-1}{\sqrt{x^2-a^2}} $ | |
| $ \int \dfrac{x^2dx}{\left(x^2-a^2\right)^{3/2}} = - \dfrac{x}{a^2\sqrt{x^2-a^2}} + \ln\left(x+\sqrt{x^2-a^2}\right) $ | |
| $ \int \dfrac{x^3dx}{\left(x^2-a^2\right)^{3/2}} = \sqrt{x^2-a^2} - \dfrac{a^2}{a^2\sqrt{x^2-a^2}} $ | |
| $ \int \dfrac{dx}{x\left(x^2-a^2\right)^{3/2}} =\dfrac{-1}{a^2 \sqrt{x^2-a^2}} - \dfrac{1}{a^3} \arccos \left|\frac{a}{x}\right \vert $ | |
| $ \int \dfrac{dx}{x^2\left(x^2-a^2\right)^{3/2}} =-\dfrac{\sqrt{x^2-a^2}}{a^4x} - \dfrac{x}{a^4\sqrt{x^2-a^2}} $ | |
| $ \int \dfrac{dx}{x^3\left(x^2-a^2\right)^{3/2}} =\dfrac{1}{2a^2x^2 \sqrt{x^2-a^2}} - \dfrac{3}{2a^4\sqrt{x^2-a^2}} - \dfrac{3}{2a^5} \arccos \left|\frac{a}{x}\right \vert $ | |
| $ \int \left(x^2-a^2\right)^{3/2} dx =\dfrac{x\left(x^2-a^2\right)^{3/2}}{4} - \dfrac{3a^2x\sqrt{x^2-a^2}}{8} + \dfrac{3}{8}a^4\ln\left(x+\sqrt{x^2-a^2}\right) $ | |
| $ \int x\left(x^2-a^2\right)^{3/2} dx =\dfrac{x\left(x^2-a^2\right)^{5/2}}{5} $ | |
| $ \int x^2\left(x^2-a^2\right)^{3/2} dx =\dfrac{x\left(x^2-a^2\right)^{5/2}}{6} + \dfrac{a^2x\left(x^2-a^2\right)^{3/2}}{24} - \dfrac{a^4x\sqrt{x^2-a^2}}{16} + \dfrac{a^6}{16}a^4\ln\left(x+\sqrt{x^2-a^2}\right) $ | |
| $ \int x^3\left(x^2-a^2\right)^{3/2} dx =\dfrac{\left(x^2-a^2\right)^{7/2}}{7} - \dfrac{a^2\left(x^2-a^2\right)^{5/2}}{5} $ | |
| $ \int \dfrac{\left(x^2-a^2 \right)^{3/2}}{x} dx = \dfrac{\left( x^2-a^2 \right)^{3/2}}{3} - a^2 \sqrt{x^2-a^2} - a^3 \arccos \left | \dfrac{a}{x} \right \vert $ | |
| $ \int \dfrac{\left(x^2-a^2\right)^{3/2}}{x^2} dx = \dfrac{x\left(x^2-a^2\right)^{3/2}}{x} - \dfrac{3x\sqrt{x^2-a^2}}{2} - \dfrac{3}{2}a^2\ln\left(x+\sqrt{x^2-a^2}\right) $ | |
| $ \int \dfrac{\left(x^2-a^2\right)^{3/2}}{x^3} dx =-\dfrac{x\left(x^2-a^2\right)^{3/2}}{2x^2} + \dfrac{3\sqrt{x^2-a^2}}{2} - \dfrac{3}{2} a \arccos \left|\frac{a}{x}\right \vert $ | |
| Particular Integral, componant $ \sqrt{a^2-x^2} $ | |
| $ \int \dfrac{dx}{\sqrt{a^2-x^2}} = \arcsin \dfrac{x}{a} $ | |
| $ \int \dfrac{xdx}{\sqrt{a^2-x^2}} = -\sqrt{a^2-x^2} $ | |
| $ \int \dfrac{x^2dx}{\sqrt{a^2-x^2}} = -\dfrac{x\sqrt{a^2-x^2}}{2} + \dfrac{a^2}{2} Arc sin \dfrac{x}{a} $ | |
| $ \int \dfrac{x^3 dx}{\sqrt{a^2-x^2}} = \dfrac{\left( a^2 - x^2 \right)^{3/2}}{3} - a^2 \sqrt{a^2-x^2} $ | |
| $ \int \dfrac{dx}{x\sqrt{a^2-x^2}} = -\dfrac{1}{a}\ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right) $ | |
| $ \int \dfrac{dx}{x^2\sqrt{a^2-x^2}} = -\dfrac{\sqrt{a^2-x^2}}{a^2x} $ | |
| $ \int \dfrac{dx}{x^3\sqrt{a^2-x^2}} = -\dfrac{\sqrt{a^2-x^2}}{2a^2x^2} -\dfrac{1}{2a^3}\ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right) $ | |
| $ \int \sqrt{a^2-x^2} dx = \dfrac{x\sqrt{a^2-x^2}}{2} - \dfrac{a^2}{2} \arcsin \dfrac{x}{a} $ | |
| $ \int x\sqrt{a^2-x^2} dx = -\dfrac{x\left(a^2-x^2\right)^{3/2}}{3} $ | |
| $ \int x^2\sqrt{a^2-x^2} dx =- \dfrac{x\left(a^2-x^2\right)^{3/2}}{4} + \dfrac{a^2x\sqrt{x^2-a^2}}{8} + \dfrac{a^4}{8} \arcsin \dfrac{a}{x} $ | |
| $ \int x^3 \sqrt{a^2-x^2} dx =\dfrac{x\left(a^2-x^2\right)^{5/2}}{5} - \dfrac{a^2\left(a^2-x^2\right)^{3/2}}{3} $ | |
| $ \int \dfrac{\sqrt{a^2 - x^2}}{x} dx = \sqrt{a^2-x^2} - a \ln \left( \dfrac{a+\sqrt{a^2-x^2}}{x} \right) $ | |
| $ \int \ dfrac{\sqrt{a^2-x^2}}{x^2} dx = - \dfrac{\sqrt{a^2-x^2}}{x} - \arcsin \dfrac{x}{a} $ | |
| $ \int \dfrac{\sqrt{a^2-x^2}}{x^3} dx = -\dfrac{\sqrt{a^2-x^2}}{2x^2} + \dfrac{1}{2a}\ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right) $ | |
| $ \int \dfrac{dx}{\left(a^2-x^2\right)^{3/2}} = \dfrac{x}{a^2\sqrt{a^2-x^2}} $ | |
| $ \int \dfrac{xdx}{\left(a^2-x^2\right)^{3/2}} = \dfrac{1}{\sqrt{a^2-x^2}} $ | |
| $ \int \dfrac{x^2 dx}{\left(a^2-x^2 \right)^{3/2}} = \dfrac{x}{\sqrt{a^2-x^2}} - \arcsin \dfrac{x}{a} $ | |
| $ \int \dfrac{x^3dx}{\left(a^2-x^2\right)^{3/2}} = \sqrt{a^2-x^2} + \dfrac{a^2}{\sqrt{a^2-x^2}} $ | |
| $ \int \dfrac{dx}{x\left(a^2-x^2\right)^{3/2}} = \dfrac{1}{a^2\sqrt{a^2-x^2}} - \dfrac{1}{a^3}\ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right) $ | |
| $ \int \dfrac{dx}{x^2\left(a^2-x^2\right)^{3/2}} = -\dfrac{\sqrt{a^2-x^2}}{a^4x} + \dfrac{x}{a^4\sqrt{a^2-x^2}} $ | |
| $ \int \dfrac{dx}{x^3\left(a^2-x^2\right)^{3/2}} = \dfrac{-1}{2a^2x^2\sqrt{a^2-x^2}} + \dfrac{3}{2a^4\sqrt{a^2-x^2}} - \dfrac{3}{2a^5}\ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right) $ | |
| $ \int \left(a^2-x^2\right)^{3/2} dx = \dfrac{x\left(a^2-x^2\right)^{3/2}}{4} + \dfrac{3a^2x\sqrt{a^2-x^2}}{8} + \dfrac{3}{8} a^4 \arcsin \dfrac {x}{a} $ | |
| $ \int x\left(a^2-x^2\right)^{3/2} dx = -\dfrac{x\left(a^2-x^2\right)^{5/2}}{5} $ | |
| $ \int x^2\left(a^2-x^2\right)^{3/2} dx = -\dfrac{x\left(a^2-x^2\right)^{5/2}}{6} + \dfrac{a^2x\left(a^2-x^2\right)^{3/2}}{24} + \dfrac{a^4x\sqrt{a^2-x^2}}{16} + \dfrac{a^6}{16} \arcsin \dfrac {x}{a} $ | |
| $ \int x^3\left(a^2-x^2\right)^{3/2} dx = \dfrac{\left(a^2-x^2\right)^{7/2}}{7} + \dfrac{a^2\left(a^2-x^2\right)^{5/2}}{5} $ | |
| $ \int \dfrac{\left(a^2-x^2\right)^{3/2}}{x} dx = \dfrac{\left(a^2-x^2\right)^{3/2}}{3} + a^2\sqrt{a^2-x^2} - a^3 \ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right) $ | |
| $ \int \dfrac{\left(a^2-x^2\right)^{3/2}}{x^2} dx = -\dfrac{\left(a^2-x^2\right)^{3/2}}{x} - \dfrac{3x\sqrt{a^2-x^2}}{2} - \dfrac{3}{2}a^2 \arcsin \dfrac{x}{a} $ | |
| $ \int \dfrac{\left(a^2-x^2\right)^{3/2}}{x^3} dx = -\dfrac{\left(a^2-x^2\right)^{3/2}}{2x^2} + \dfrac{3\sqrt{a^2-x^2}}{2} + \dfrac{3}{2}a\ln\left(\dfrac{a+\sqrt{a^2-x^2}}{x}\right) $ | |
| Particular Integral, componant $ ax^2 + bx + c $ | |
| $ \int \dfrac{dx}{ax^2 + bx + c} = \begin{cases} \dfrac{2}{\sqrt{4ac-b^2}} \arctan \dfrac{2ax+b}{\sqrt{4ac-b^2}} \\ \dfrac{1}{\sqrt{b^2-4ac}} \ln\left(\dfrac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right) \end{cases} $ | |
| $ Si\quad b^2 = 4ac, ax^2 + bx + c = a\left(x+ b/2a\right)^2 et\quad on\quad peut\quad utiliser\quad les\quad r\acute{e}sultats\quad des\quad pages \quad60-61. $ | |
| $ \quad Si \quad b = 0 \quad utiliser \quad les \quad r\acute{e}sultats \quad de \quad la \quad page \quad 64. \quad Si \quad a \quad ou \quad c = 0,\quad utiliser \quad les $ | |
| $ \quad r\acute{e}sultats \quad des \quad pages \quad 60-61. $ | |
| $ \int \dfrac{xdx}{ax^2 + bx + c} = \dfrac{1}{2a} \ln\left(ax^2+bx+c\right) - \dfrac{b}{2a}\int \dfrac{dx}{ax^2 + bx + c} $ | |
| $ \int \dfrac{x^2dx}{ax^2 + bx + c} = \dfrac{x}{a} - \dfrac{b}{2a^2}\ln\left(ax^2+bx+c\right) + \dfrac{b^2-2ac}{2a^2} \dfrac{b}{2a}\int \dfrac{dx}{ax^2 + bx + c} $ | |
| $ \int \dfrac{x^{m}dx}{ax^2 + bx + c} = \dfrac{x^{m-1}}{\left(m-1\right)a} - \dfrac{c}{a}\int \dfrac{x^{m-2}dx}{ax^2 + bx + c} - \dfrac{b}{a}\int \dfrac{x{m-1}dx}{ax^2 + bx + c} $ | |
| $ \int \dfrac{dx}{x\left(ax^2 + bx + c\right)} = \dfrac{1}{2c} \ln\left(\dfrac{x^2}{ax^2+bx+c}\right) - \dfrac{b}{2c} \int \dfrac{dx}{ax^2+bx+c} $ | |
| $ \int \dfrac{dx}{x^2\left(ax^2 + bx + c\right)} = \dfrac{b}{2c^2} \ln\left(\dfrac{ax^2+bx+c}{x^2}\right) - \dfrac{1}{cx} + \dfrac{b^2-2ac}{2c^2} \int \dfrac{dx}{ax^2+bx+c} $ | |
| $ \int \dfrac{dx}{x^n\left(ax^2 + bx + c\right)} = - \dfrac{1}{\left(n-1\right)cx^{n-1}} - \dfrac{b}{c}\int \dfrac{dx}{x^{n-1}\left(ax^2+bx+c\right)} - \dfrac{a}{c}\int \dfrac{dx}{x^{n-2}\left(ax^2+bx+c\right)} $ | |
| $ \int \dfrac{dx}{\left(ax^2+bx+c \right)^2} = \dfrac{2ax+b}{\left(4ac-b^2 \right) \left(ax^2+bx+c \right)} + \dfrac{2a}{4ac-b^2} \int \dfrac{dx}{ax^2+bx+c} $ | |
| $ \int \dfrac{xdx}{\left(ax^2+bx+c^2 \right)^2} = - \dfrac{bx+2c}{\left(4ac-b^2\right)\left(ax^2+bx+c \right)} - \dfrac{b}{4ac-b^2} \int \dfrac{dx}{ax^2+bx+c} $ | |
| $ \int \dfrac{x^2 dx}{\left(ax^2+bx+c \right)^2} = \dfrac{\left(b^2-2ac\right)x+bc}{a\left(4ac-b^2\right)\left(ax^2+bx+c\right)} + \dfrac{2c}{4ac-b^2} \int \dfrac{dx}{ax^2+bx+c} $ | |
| $ \int \dfrac{x^{m}dx}{\left(ax^2+bx+c\right)^n} = - \dfrac{x^{m-1}}{\left(2n-m-1\right)a\left(ax^2+bx+c\right)^{n-1}} + \dfrac{\left(m-1\right)c}{\left(2n-m-1\right)a} \int \dfrac{x^{m-2}dx}{\left(ax^2+bx+c\right)^{n}} $ | |
| $ - \dfrac{\left(n-m\right)b}{\left(2n-m-1\right)a} \int \dfrac{x^{m-1}dx}{\left(ax^2+bx+c\right)^n} $ | |
| $ \int \dfrac{x^{2n-1}dx}{\left(ax^2+bx+c\right)^{n}} = \dfrac{1}{a} \int \dfrac{x^{2n-3}dx}{\left(ax^2+bx+c\right)^{n-1}} - \dfrac{c}{a} \int \dfrac{x^{2n-3}dx}{\left(ax^2+bx+c\right)^{n}} - \dfrac{b}{a} \int \dfrac{x^{2n-2}dx}{\left(ax^2+bx+c\right)^n} $ | |
| $ \int \dfrac{dx}{x\left(ax^2+bx+c\right)^2} = \dfrac{1}{2c\left(ax^2+bx+c\right)} - \dfrac{b}{2c} \int \dfrac{dx}{\left(ax^2+bx+c\right)^2} + \dfrac{1}{c} \int \dfrac{dx}{x\left(ac^2+bx+c\right)} $ | |
| $ \int \dfrac{dx}{x^2\left(ax^2+bx+c\right)^2} = -\dfrac{1}{cx\left(ax^2+bx+c\right)} - \dfrac{3a}{c} \int \dfrac{dx}{\left(ax^2+bx+c\right)^2} - \dfrac{2b}{c} \int \dfrac{dx}{x\left(ac^2+bx+c\right)^2} $ | |
| $ \int \dfrac{dx}{x^{m}\left(ax^2+bx+c\right)^{n}} = -\dfrac{1}{\left(m-1\right)cx^{m-1}\left(ax^2+bx+c\right)^{n-1}} - \dfrac{\left(m+2n-3\right)a}{\left(m-1\right)c} \int \dfrac{dx}{x^{m-2}\left(ax^2+bx+c\right)^{n}} $ | |
| $ + \dfrac{\left(m+n-2\right)b}{\left(m-1\right)c} \int \dfrac{dx}{x^{m-1}\left(ac^2+bx+c\right)^{n}} $ | |
| Particular Integral, componant $ \sqrt{ax^2 + bx + c} $ | |
| $ Dans\quad les\quad r\acute{e}sultats\quad suivants,\quad si\quad b^2 = 4ac, \sqrt{ax^2 + bx + c} = \sqrt{a}\left(x+ b/2a\right) et\quad on\quad peut\quad utiliser\quad $ | |
| $ les\quad r\acute{e}sultats\quad des\quad pages \quad60-61.\quad Si \quad b = 0 \quad utiliser \quad les \quad r\acute{e}sultats \quad de \quad la \quad page \quad 64. $ | |
| $ Si \quad a \quad ou \quad c = 0,\quad utiliser \quad les\quad r\acute{e}sultats \quad des \quad pages \quad 60-61. $ | |
| $ \int \dfrac{dx}{\sqrt{ax^2+bx+c}} = \begin{cases} \dfrac{1}{\sqrt{a}} \ln\left(2\sqrt{a}\sqrt{ax^2+bx+c}+ax+b\right)\\ -\dfrac{1}{\sqrt{-a}} \arcsin\left(\dfrac{2ax+b}{\sqrt{b^2-4ac}}\right)\quad ou\quad \dfrac{1}{\sqrt{a}} argsh\left(\dfrac{2ax+b}{\sqrt{4ac-b^2}}\right) \end{cases} $ | |
| $ \int \dfrac{xdx}{\sqrt{ax^2+bx+c}} = \dfrac{\sqrt{ax^2+bx+c}}{a} - \dfrac{b}{2a} \int \dfrac{dx}{\sqrt{ax^2+bx+c}} $ | |
| $ \int \dfrac{x^2dx}{\sqrt{ax^2+bx+c}} = \dfrac{2ax-3b}{4a^2}\sqrt{ax^2+bx+c}+\dfrac{3b^2-4ac}{8a^2}\int\dfrac{dx}{\sqrt{ax^2+bx+c}} $ | |
| $ \int \dfrac{dx}{x\sqrt{ax^2+bx+c}} = \begin{cases} -\dfrac{1}{\sqrt{c}}\ln\left(\dfrac{2\sqrt{c}\sqrt{ax^2+bx+c}+bx+2c}{x}\right)\\ -\dfrac{1}{\sqrt{-c}} \arcsin\left(\dfrac{bx+2c}{\left|x\right\vert\sqrt{b^2-4ac}}\right)\quad ou\quad -\dfrac{1}{\sqrt{c}}argsh\left(\dfrac{bx+2c}{\left|x\right\vert\sqrt{4ac-b^2}}\right)\end{cases} $ | |
| $ \int \dfrac{dx}{x^2\sqrt{ax^2+bx+c}} = -\dfrac{\sqrt{ax^2+bx+c}}{cx} - \dfrac{b}{2c}\int\dfrac{dx}{x\sqrt{ax^2+bx+c}} $ | |
| $ \int \sqrt{ax^2+bx+c} dx = \dfrac{\left(2ax+b\right)\sqrt{ax^2+bx+c}}{4a} + \dfrac{4ac-b^2}{8a}\int\dfrac{dx}{\sqrt{ax^2+bx+c}} $ | |
| $ \int x\sqrt{ax^2+bx+c} dx = \dfrac{\left(ax^2+bx+c\right)^{3/2}}{3a} - \dfrac{b\left(2ax+b\right)}{8a^2}\sqrt{ax^2+bx+c} - \dfrac{b\left(4ac-b^2\right)}{16a^2} \int \dfrac{dx}{\sqrt{ax^2+bx+c}} $ | |
| $ \int x^2\sqrt{ax^2+bx+c} dx = \dfrac{6ax-5b}{24a^2} \left(ax^2+bx+c\right)^{3/2} + \dfrac{5b^2-4ac}{16a^2}\int\sqrt{ax^2+bx+c} dx $ | |
| $ \int \dfrac{\sqrt{ax^2+bx+c}}{x} dx = \sqrt{ax^2+bx+c} + \dfrac{b}{2} \int\dfrac{dx}{\sqrt{ax^2+bx+c}} + c\int\dfrac{dx}{x\sqrt{ax^2+bx+c}} $ | |
| $ \int \dfrac{\sqrt{ax^2+bx+c}}{x^2} dx = -\dfrac{\sqrt{ax^2+bx+c}}{x} + a\int\dfrac{dx}{\sqrt{ax^2+bx+c}} + \dfrac{b}{2}\int\dfrac{dx}{x\sqrt{ax^2+bx+c}} $ | |
| $ \int \dfrac{dx}{\left(ax^2+bx+c\right)^{3/2}} = \dfrac{2\left(2ax+b\right)}{\left(4ac-b^2\right)\sqrt{ax^2+bx+c}} $ | |
| $ \int \dfrac{xdx}{\left(ax^2+bx+c\right)^{3/2}} = \dfrac{2\left(2bx+2c\right)}{\left(b^2-4ac\right)\sqrt{ax^2+bx+c}} $ | |
| $ \int \dfrac{x^2dx}{\left(ax^2+bx+c\right)^{3/2}} = \dfrac{\left(2b^2-4ac\right)x+2bc}{a\left(4ac-b^2\right)\sqrt{ax^2+bx+c}}+\dfrac{1}{a}\int\dfrac{dx}{\sqrt{ax^2+bx+c}} $ | |
| $ \int\dfrac{dx}{x\left(ax^2+bx+c\right)^{3/2}} = \dfrac{1}{c\sqrt{ax^2+bx+c}} + \dfrac{1}{c}\int\dfrac{dx}{x\sqrt{ax^2+x+c}} - \dfrac{b}{2c}\int\dfrac{dx}{\left(ax^2+bx+c\right)^{3/2}} $ | |
| $ \int\dfrac{dx}{x^2\left(ax^2+bx+c\right)^{3/2}} = -\dfrac{ax^2+2bx+c}{c^2x\sqrt{ax^2+bx+c}} + \dfrac{v^2-2ac}{2c^2}\int\dfrac{dx}{\left(ax^2+x+c\right)^{3/2}} - \dfrac{3b}{2c^2}\int\dfrac{dx}{x\sqrt{ax^2+bx+c}} $ | |
| $ \int\left(ax^2+bx+c\right)^{n+1/2} dx = \dfrac{\left(2ax+b\right)\left(ax^2+bx+c\right)^{n+1/2}}{4a\left(n+1\right)} + \dfrac{\left(2n+1\right)\left(4ac-b^2\right)}{8a\left(n+1\right)}\int\left(ax^2+bx+c\right)^{n-1/2}dx $ | |
| $ \int x\left(ax^2+bx+c\right)^{n+1/2}dx = \dfrac{\left(ax^2+bx+c\right)^{n+3/2}}{a\left(2n+3\right)} - \dfrac{b}{2a}\int \left((ax^2+bx+c\right)^{n+1/2} dx $ | |
| $ \int \dfrac{dx}{\left(ax^2+bx+c\right)^{n+1/2}} = \dfrac{2\left(2ax+b\right)}{\left(2n-1\right)\left(4ac-b^2\right)\left(ax^2+bx+c\right)^{n-1/2}} + \dfrac{8a\left(n-1\right)}{\left(2n-1\right)\left(4ac-b^2\right)} \int \dfrac{dx}{\left(ax^2+bx+c\right)^{n-1/2}} $ | |
| $ \int \dfrac{dx}{x\left(ax^2+bx+c\right)^{n+1/2}} = \dfrac{1}{\left(2n-1\right)c\left(ax^2+bx+c\right)^{n-1/2}} + \dfrac{1}{c} \int \dfrac{dx}{x\left(ax^2+bx+c\right)^{n-1/2}} - \dfrac{b}{2c} \int \dfrac{dx}{\left(ax^2+bx+c\right)^{n+1/2}} $ | |
| Particular Integral, componant $ x^3+a^3 $ | |
| $ \int \dfrac{dx}{x^3+a^3} = \dfrac{1}{6a^2} \ln \dfrac{\left(x+a\right)^2}{x^2-ax+a^2} + \dfrac{1}{a^2\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}} $ | |
| $ \dfrac{xdx}{x^3+a^3} = \dfrac{1}{6a}\ln \dfrac{x^2-ax+a^2}{\left(x+a\right)^2} + \dfrac{1}{a\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}} $ | |
| $ \int \dfrac{x^2dx}{x^3+a^3} = \dfrac{1}{3} \ln\left(x^3+a^3\right) $ | |
| $ \int \dfrac{dx}{x\left(x^3+a^3\right)} = \dfrac{1}{3a^3} \ln \left(\dfrac{x^3}{x^3+a^3}\right) $ | |
| $ \int \dfrac{dx}{x^2\left(x^3+a^3\right)} = -\dfrac{1}{a^3x} - \dfrac{1}{6a^4} \ln \dfrac{x^2-ax+a^2}{\left(x+a\right)} - \dfrac{1}{a^4\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}} $ | |
| $ \int \dfrac{dx}{\left(x^3+a^3\right)^2} = \dfrac{x}{3a^3\left(x^3+a^3\right)} + \dfrac{1}{9a^5} \ln\dfrac{\left(x+a\right)^2}{x^2-ax+a^2} + \dfrac{2}{3a^5\sqrt{3}}\arctan\dfrac{2x-a}{a\sqrt{3}} $ | |
| $ \int \dfrac{xdx}{\left(x^3+a^3\right)^2} = \dfrac{x^2}{3a^3\left(x^3+a^3\right)} + \dfrac{1}{18a^4} \ln \dfrac{x^2-ax+a^2}{\left(x+a\right)^2} + \dfrac{1}{3a^4\sqrt{3}} \arctan \dfrac{2x-a}{a\sqrt{3}} $ | |
| $ \int \dfrac{x^2dx}{\left(x^3+a^3\right)} = -\dfrac{1}{3\left(x^3+a^3\right)} $ | |
| $ \int \dfrac{dx}{x\left(x^3+a^3\right)} = \dfrac{1}{3a^3\left(x^3+a^3\right)} + \dfrac{1}{3a^6}\ln\left(\dfrac{x^3}{x^3+a^3}\right) $ | |
| $ \int \dfrac{dx}{x^2\left(x^3+a^3\right)} = -\dfrac{1}{a^6x} - \dfrac{x^2}{3a^6\left(x^3+a^3\right)} - \dfrac{4}{3a^6}\int \dfrac{xdx}{x^3+a^3} $ | |
| $ \int \dfrac{x^{m}dx}{x^3+a^3} = \dfrac{x^{m-2}}{m-2} - a^3 \int \dfrac{x^{m-3}dx}{x^3+a^3} $ | |
| $ \int \dfrac{dx}{x^{n}\left(x^3+a^3\right)} = \dfrac{-1}{a^3\left(n-1\right)x^{n-1}} - \dfrac{1}{a^3} \int \dfrac{dx}{x^{n-3}\left(x^3+a^3\right)} $ | |
| Particular Integral, componant $ x^4 \pm a^4 $ | |
| $ \int \dfrac{dx}{x^4+a^4} = \dfrac{1}{4a^3\sqrt{2}}\ln\left(\dfrac{x^2+ax\sqrt{2}+a^2}{x^2-ax\sqrt{2}+a^2}\right) - \dfrac{1}{2a^3\sqrt{2}}\arctan \dfrac{ax\sqrt{2}}{x^2-a^2} $ | |
| $ \int \dfrac{xdx}{x^4+a^4} = \dfrac{1}{2a^2} \arctan \dfrac{x^2}{a^2} $ | |
| $ \int \dfrac{x^2da}{x^4+a^4} = \dfrac{1}{4a\sqrt{2}} \ln\left(\dfrac{x^2-ax\sqrt{2}+a^2}{x^2+ax\sqrt{2}+a^2}\right) = \dfrac{1}{2a\sqrt{2}} \arctan \dfrac{ax\sqrt{2}}{x^2-a^2} $ | |
| $ \int \dfrac{x^3dx}{x^4+a^4} = \dfrac{1}{4} \ln \left(x^4+a^4\right) $ | |
| $ \int \dfrac{dx}{x\left(x^4+a^4\right)} = \dfrac{1}{4a^4}\ln\left(\dfrac{x^4}{x^4+a^4}\right) $ | |
| $ \int \dfrac{dx}{x^2\left(x^4+a^4\right)} = -\dfrac{1}{a^4x} - \dfrac{1}{4a^5\sqrt{2}}\ln\left(\dfrac{x^2-ax\sqrt{2}+a^2}{x^2+ax\sqrt{2}+a^2}\right) + \dfrac{1}{2a^5\sqrt{2}}\arctan\dfrac{ax\sqrt{2}}{x^2-a^2} $ | |
| $ \int \dfrac{dx}{x^3\left(x^4+a^4\right)} = -\dfrac{1}{2a^4x^2} - \dfrac{1}{a^6} \arctan\dfrac{x^2}{a^2} $ | |
| $ \int \dfrac{dx}{x^4-a^4} = \dfrac{1}{4a^3}\ln\left(\dfrac{x-a}{x+a}\right) - \dfrac{1}{2a^3}\arctan\dfrac{x}{a} $ | |
| $ \int \dfrac{xdx}{x^4-a^4} = \dfrac{1}{4a^2}\ln\left(\dfrac{x^2-a^2}{x^2+a^2}\right) $ | |
| $ \int \dfrac{x^2dx}{x^4-a^4} = \dfrac{1}{4a}\ln\left(\dfrac{x-a}{x+a}\right) + \dfrac{1}{2a}\arctan\dfrac{x}{a} $ | |
| $ \int \dfrac{x^3dx}{x^4-a^4} = \dfrac{1}{4} \ln\left(x^4-a^4\right) $ | |
| $ \int \dfrac{dx}{x\left(x^3-a^4\right)} = \dfrac{1}{4a^4}\ln\left(\dfrac{x^4-a^4}{x^4}\right) $ | |
| $ \int \dfrac{dx}{x^2\left(x^4-a^4\right)} = \dfrac{1}{a^4x} + \dfrac{1}{4a^5}\ln\left(\dfrac{x-a}{x+a}\right) + \dfrac{1}{2a^5}\arctan\dfrac{x}{a} $ | |
| $ \int \dfrac{dx}{x^3\left(x^4-a^4\right)} = \dfrac{1}{2a^4x^2} + \dfrac{1}{4a^6}\ln\left(\dfrac{x^2-a^2}{x^2+a^2}\right) $ | |
| Particular Integral, componant $ x^{n} + a^{n} $ | |
| $ \int \dfrac{dx}{x\left(x^n+a^n\right)} = \dfrac{1}{na^{n}}\ln\dfrac{x^{n}}{x^{n}+a^{n}} $ | |
| $ \int \dfrac{x^{n-1}}{x^{n}+a^{n}} = \dfrac{1}{n}\ln\left(x^{n}+a^{n}\right) $ | |
| $ \int \dfrac{x^{m}}{\left(x^{n}+a^{n}\right)^{r}} = \int \dfrac{x^{m-n}dx}{\left(x^{n}+a^{n}\right)^{r-1}} - a^{n}\int\dfrac{x^{m-n}dx}{\left(x^{n}+a^{n}\right)^{r}} $ | |
| $ \int \dfrac{dx}{x^{m}\left(x^{n}+a^{n}\right)^{r}} = \dfrac{1}{a^{n}}\int\dfrac{dx}{x^{m}\left(x^{n}+a{n}\right)^{r-1}} - \dfrac{1}{a^{n}}\int\dfrac{dx}{x^{m-n}\left(x^{n}+a^{n}\right)^{r}} $ | |
| $ \int \dfrac{dx}{x\sqrt{x^{n}+a^{n}}} = \dfrac{1}{n\sqrt{a^{n}}}\ln\left(\dfrac{\sqrt{x^{n}+a^{n}}-\sqrt{a^{n}}}{\sqrt{x^{n}+a^{n}}+\sqrt{a^{n}}}\right) $ | |
| $ \int \dfrac{dx}{x\left(x^{n}-a^{n}\right)} = \dfrac{1}{na^{n}}\ln\left(\dfrac{x^{n}-a^{n}}{x^{n}}\right) $ | |
| $ \int \dfrac{x^{n-1}dx}{x^{n}-a^{n}} = \dfrac{1}{n}\ln\left(x^{n}-a^{n}\right) $ | |
| $ \int \dfrac{x^{m}dx}{\left(x^{n}-a^{n}\right)^{r}} = a^{n}\int\dfrac{x^{m-n}dx}{\left(x^{n}-a^{n}\right)^{r}} + \int\dfrac{x^{m-n}dx}{\left(x^{n}-a^{n}\right)^{r-1}} $ | |
| $ \int\dfrac{dx}{x^{m}\left(x^{n}-a^{n}\right)^{r}} = \dfrac{1}{a^{n}}\int\dfrac{dx}{x^{m-n}\left(x^{n}-a^{n}\right)^{r}}-\dfrac{1}{a^{n}}\int\dfrac{dx}{x^{m}\left(x^{n}-a^{n}\right)^{r-1}} $ | |
| $ \int\dfrac{dx}{x\sqrt{x^{n}-a^{n}}} = \dfrac{2}{n\sqrt{a^{n}}}\arccos\sqrt{\dfrac{a^{n}}{x^{n}}} $ | |
| $ \int \dfrac{x^{p-1}dx}{x^{2m}+a^{2m}} = \dfrac{1}{ma^{2m-p}} \sum_{k=1}^m \sin\dfrac{\left(2k-1\right)p\pi}{2m}\arctan\left(\dfrac{x+a\cos\left[\left(2k-1\right)\pi /2m\right]}{a\sin\left[\left(2k-1\right)\pi /2m\right]}\right) - \dfrac{1}{2ma^{2m-p}} \sum_{k=1}^m \cos\dfrac{\left(2k-1\right)p\pi}{2m}\ln\left(x^2+2ax\cos\dfrac{\left(2k-1\right)\pi}{2m} + a^2\right) $ | |
| $ \int \dfrac{x^{p-1}dx}{x^{2m}-a^{2m}} = \dfrac{1}{2ma^{2m-p}} \sum_{k=1}^{m-1} \cos\dfrac{kp\pi}{m}\ln\left(x^2 - 2ax\cos\dfrac{k\pi}{m} + a^2\right) - \dfrac{1}{ma^{2m-p}} \sum_{k=1}^{m-1} \sin \dfrac{kp\pi}{m} \arctan\left(\dfrac{x-a\cos\left(k\pi /m\right)}{a\sin\left(k\pi/m\right)}\right) + \dfrac{1}{2ma^{2m-p}}\left\{\ln\left(x-a\right)+\left(-1\right)^{p}\ln\left(x+a\right)\right\} $ | |
| $ \int \dfrac{x^{p-1}dx}{x^{2m+1}+a^{2m+1}} = \dfrac{2\left(-1\right)^{p-1}}{\left(2m+1\right)a^{2m-p+1}}\sum_{k=1}^m\sin\dfrac{2kp\pi}{2m+1}\arctan\left(\dfrac{x+a\cos\left[2k\pi/ \left(2m+1\right)\right]}{a\sin\left[2k\pi/ \left(2m+1\right)\right]}\right)-\dfrac{\left(-1\right)^{p-1}}{\left(2m+1\right)a^{2m-p+1}}\sum_{k=1}^m \cos\dfrac{2kp\pi}{2m+1}\ln\left(x^2+2ax\cos\dfrac{2k\pi}{2m+1}+a^2\right) + \dfrac{\left(-1\right)^{p-1}}{\left(2m+1\right)a^{2m-p+1}} $ | |
| $ \int \dfrac{x^{p-1}dx}{x^{2m+1}-a^{2m+1}} = \dfrac{-2}{\left(2m+1\right)a^{2m-p+1}} \sum_{k=1}^m \sin \dfrac{2kp\pi}{2m+1}\arctan\left(\dfrac{x-a\cos\left[2k\pi/\left(2m+1\right)\right]}{a\sin\left[2k\pi/\left(2m+1\right)\right]}\right) + \dfrac{1}{\left(2m+1\right)a^{2m-p+1}}\sum_{k=1}^m \cos\dfrac{2kp\pi}{2m+1}\ln\left(x^2-2ax\cos\dfrac{2k\pi}{2m+1}+a^2\right) + \dfrac{\ln\left(x-a\right)}{\left(2m+1\right)a^{2m-p+1}} $ | |
