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| + | [[Category:Formulas]] | ||
| + | [[Category:integral]] | ||
| + | |||
| + | <center><font size= 4> | ||
| + | '''[[Collective_Table_of_Formulas|Collective Table of Formulas]]''' | ||
| + | </font size> | ||
| + | |||
| + | '''Indefinite Integrals''' | ||
| + | |||
| + | click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]] | ||
| + | |||
| + | </center> | ||
| + | |||
| + | ---- | ||
{| | {| | ||
|- | |- | ||
| − | + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | [[Table_of_indefinite_integrals_general_rules|General Rules]] | |
| − | + | ||
| − | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | General Rules | + | |
|- | |- | ||
|<math> \int a d x = a x </math> | |<math> \int a d x = a x </math> | ||
| Line 15: | Line 27: | ||
|<math> \int f ( a x ) d x = \frac{1}{a} \int f ( u ) d u </math> | |<math> \int f ( a x ) d x = \frac{1}{a} \int f ( u ) d u </math> | ||
|- | |- | ||
| − | |<math> \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 ) </math> | + | |<math> \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 ) </math> |
|- | |- | ||
|<math> \int u^n d u = \frac{u^{n+1}}{n+1} \qquad n \neq -1 </math> | |<math> \int u^n d u = \frac{u^{n+1}}{n+1} \qquad n \neq -1 </math> | ||
|- | |- | ||
| − | |<math> \int \frac{d u}{u} = \ln u \ ( if \ u > 0 ) \ or \ln {-u} \ ( if \ u < 0 ) = \ln \left | u \right | </math> | + | |<math> \int \frac{d u}{u} = \ln u \ ( if \ u > 0 ) \ \text{or} \ln {-u} \ ( \text{if} \ u < 0 ) = \ln \left | u \right | </math> |
|- | |- | ||
|<math> \int e^u d u = e^u </math> | |<math> \int e^u d u = e^u </math> | ||
|- | |- | ||
| − | |< | + | |<font size= 4> Click [[Table_of_indefinite_integrals_general_rules|here]] for [[Table_of_indefinite_integrals_general_rules|more general rules]]. </font size> |
|- | |- | ||
| − | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | | + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Transformations of the independent variable |
|- | |- | ||
| − | |<math> \int a d x = a x </math> | + | |<math> \int F( a x + b) d x =\frac{1}{a} \int F( u) d u \qquad u = a x + b</math> |
| + | |- | ||
| + | |<math> \int F( \sqrt {a x + b} ) d x =\frac{2}{a} \int u F( u) d u \qquad u = \sqrt {a x + b} </math> | ||
| + | |- | ||
| + | |<math> \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} </math> | ||
| + | |- | ||
| + | |<math> \int F( \sqrt {a^2 - x^2} ) d x =a \ \int F( a \cos u) \ \cos u \ d u \qquad x = a \sin u </math> | ||
| + | |- | ||
| + | |<math> \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 </math> | ||
| + | |- | ||
| + | |<math> \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} </math> | ||
| + | |- | ||
| + | |<math> \int F( e ^{a x}) d x = \frac {1}{a} \int \frac {F(u)}{u} \ d u \qquad u = e^{a x}</math> | ||
| + | |- | ||
| + | |<math> \int F( \ln x ) d x = \int F(u)\ e^u \ d u \qquad u = \ln x</math> | ||
| + | |- | ||
| + | |<math> \int F\left ( \arcsin \frac{x}{a} \right) d x = a \int F(u)\ \cos u \ d u \qquad u = \arcsin \frac {x}{a}</math> | ||
| + | |- | ||
| + | |<math> \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}</math> | ||
| + | |- | ||
| + | |- | ||
| + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | [[Table_of_indefinite_integrals_axplusb|Integrals with ax +b]] | ||
| + | |- | ||
| + | |<math> \int \frac {d x}{ ax + b} = \frac {1}{a} \ln (ax +b)</math> | ||
| + | |- | ||
| + | |<math> \int \frac {x d x}{ ax + b} = \frac {x}{a} - \frac{b}{a^2} \ln (ax +b)</math> | ||
| + | |- | ||
| + | |<math> \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)</math> | ||
| + | |- | ||
| + | |<math> \int \frac {d x}{\sqrt{a x +b}} = \frac {2\sqrt{ax+b}}{a}</math> | ||
| + | |- | ||
| + | |<math> \int \frac {x d x}{\sqrt{a x + b}} = \frac {2(ax-2b)}{3a^2}\sqrt{ax+b}</math> | ||
| + | |- | ||
| + | |<math> \int \frac {x^2 d x}{\sqrt{a x + b}} = \frac {2(3a^2x^2-4abx + 8b^2)}{15a^3}\sqrt{ax+b}</math> | ||
|- | |- | ||
| + | |<font size= 4> Click [[Table_of_indefinite_integrals_axplusb|here]] for [[Table_of_indefinite_integrals_axplusb|more integrals with ax+b]]. </font size> | ||
| + | |} | ||
| + | ---- | ||
| + | ==More== | ||
| + | * [[Table_of_indefinite_integrals_axplusb_and_pxplusq|Integrals with ax+b and px+q]]. | ||
| + | * [[Table_of_indefinite_integrals_xnplusan|Integrals with <math>x^n+a^n</math>]] | ||
| + | * [[Table_of_indefinite_integrals_xnminusan|Integrals with <math>x^n-a^n</math>]] | ||
| + | * [[Table_of_indefinite_integrals_cosine_sine|Integrals with <math>\cos x</math> and/or <math>\sin x</math>]] | ||
| + | * [[Table_of_indefinite_integrals_cosine_sine|Integrals with <math>\cos x</math> and/or <math>\sin x</math>]] | ||
| + | *[[Table_of_indefinite_integrals_cotangent|Integrals with cotangent (cot x)]] | ||
| + | *[[Table_of_indefinite_integrals_oneovercosine|Integrals with 1/cos x]] | ||
| + | *[[Table_of_indefinite_integrals_inversecircularfunctions|Integrals with arccos, arcsin, arctan, arc cot]] | ||
| + | *[[Table_of_indefinite_integrals_exponential|Integrals with <math>e^x</math>]] | ||
| + | *[[Table_of_indefinite_integrals_log|Integrals with <math>\ln x</math>]] | ||
| + | *[[Table_of_indefinite_integrals_sh|Integrals with hyperbolic sine (sh x)]] | ||
| + | *[[Table_of_indefinite_integrals_ch|Integrals with hyperbolic cosine (ch x)]] | ||
| + | *[[Table_of_indefinite_integrals_th|Integrals with hyperbolic tangent (th x)]] | ||
| + | *[[Table_of_indefinite_integrals_coth|Integrals with hyperbolic cotangent (coth x)]] | ||
| + | *[[Table_of_indefinite_integrals_x_inverse|Integrals with 1/x]] | ||
| + | *[[Table_of_indefinite_quadratic|Integrals with <math>ax^2+bx+c</math>]] | ||
| + | |||
| + | ---- | ||
| + | [[Collective_Table_of_Formulas|Back to Collective Table of Formulas]] | ||
Latest revision as of 18:06, 26 February 2015
Indefinite Integrals
click here for more formulas
| 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 $ | |
| Click here for more general rules. | |
| Transformations of the independent variable | |
| $ \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} $ | |
| Integrals with 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 {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} $ | |
| Click here for more integrals with ax+b. | |
More
- Integrals with ax+b and px+q.
- Integrals with $ x^n+a^n $
- Integrals with $ x^n-a^n $
- Integrals with $ \cos x $ and/or $ \sin x $
- Integrals with $ \cos x $ and/or $ \sin x $
- Integrals with cotangent (cot x)
- Integrals with 1/cos x
- Integrals with arccos, arcsin, arctan, arc cot
- Integrals with $ e^x $
- Integrals with $ \ln x $
- Integrals with hyperbolic sine (sh x)
- Integrals with hyperbolic cosine (ch x)
- Integrals with hyperbolic tangent (th x)
- Integrals with hyperbolic cotangent (coth x)
- Integrals with 1/x
- Integrals with $ ax^2+bx+c $
