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| <math> \int\dfrac{dx}{sh ax(ch ax-1)}=-\dfrac{1}{2a}\ln th\dfrac{ax}{2}-\dfrac{1}{2a(ch ax-1)} +C</math> | | <math> \int\dfrac{dx}{sh ax(ch ax-1)}=-\dfrac{1}{2a}\ln th\dfrac{ax}{2}-\dfrac{1}{2a(ch ax-1)} +C</math> | ||
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| + | ! 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" | Inverse Hyperbolic Sine ( arg sh x) | ||
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| + | | <math> \int\arg sh\dfrac{x}{a}dx=x\arg sh\dfrac{x}{a}-\sqrt{x^{2}+a^{2}} +C</math> | ||
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| + | | <math> \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} +C</math> | ||
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| + | | <math> \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} +C</math> | ||
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| + | | <math> \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} +C</math> | ||
| + | |- | ||
| + | | <math> \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) +C</math> | ||
|} | |} | ||
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Latest revision as of 17:42, 26 February 2015
Indefinite Integrals with hyperbolic sine (sh x)
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| $ \int sh ax dx=\dfrac{ch ax}{a} +C $ | |
| $ \int x sh ax dx=\dfrac{x ch ax}{a}-\dfrac{sh ax}{a^{2}} +C $ | |
| $ \int x^{2} sh ax dx=(\dfrac{x^{2}}{a^{2}}+\dfrac{2}{a^{3}}) ch ax-\dfrac{2x}{a^{2}} sh ax +C $ | |
| $ \int\dfrac{sh ax}{x} dx=ax+\dfrac{(ax)^{3}}{3\cdot3!}+\dfrac{(ax)^{5}}{5\cdot5!}+\cdots +C $ | |
| $ \int\dfrac{sh ax}{x^{2}} dx=- \dfrac{sh ax}{x}+a \int\dfrac{ch ax}{x}dx +C $ | |
| $ \int\dfrac{dx}{sh ax}=\dfrac{1}{a}\ln th\dfrac{ax}{2} +C $ | |
| $ \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)!}\} +C $ | |
| $ \int sh^{2} ax dx=\dfrac{sh ax ch ax}{2a}-\dfrac{x}{2} +C $ | |
| $ \int x sh^{2} ax dx=\dfrac{x sh2ax}{4a}-\dfrac{ch2ax}{8a^{2}}-\dfrac{x^{2}}{4} +C $ | |
| $ \int\dfrac{dx}{sh^{2} ax}=-\dfrac{coth ax}{a} +C $ | |
| $ \int sh ax sh px dx=\dfrac{sh(a+p) x}{2(a+p)}-\dfrac{sh(a-p)x}{2(a-p)}+C, p=\pm a $ | |
| $ \int sh ax sin px dx=\dfrac{a ch ax sin px-p sh ax cos px}{a^{2}+p^{2}} +C $ | |
| $ \int sh ax cos px dx=\dfrac{a ch ax cos px+p sh ax sin px}{a^{2}+p^{2}} +C $ | |
| $ \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}}}) +C $ | |
| $ \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) +C $ | |
| $ \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}) +C $ | |
| $ \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} $ | |
| $ \int\dfrac{1}{sh ax}dx=\dfrac{1}{a}\ln th\dfrac{ax}{2} +C $ | |
| $ \int\dfrac{1}{sh^{2} ax}dx=-\dfrac{coth ax}{a} +C $ | |
| $ \int\dfrac{1}{sh^{3} ax}dx=\dfrac{coth ax}{2a sh ax}+\dfrac{1}{2a}\ln th\dfrac{ax}{2} +C $ | |
| $ \int\dfrac{coth ax}{sh^{n} ax}dx=-\dfrac{1}{na sh^{n} ax} +C $ | |
| $ \int sh ax dx=\dfrac{ch ax}{a} +C $ | |
| $ \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\} +C $ | |
| $ \int\dfrac{xdx}{sh^{2} ax}=\dfrac{x coth ax}{a}-\dfrac{1}{a^{2}}\ln sh ax +C $ | |
| $ \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\} +C $ | |
| $ \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} $ | |
| $ \int sh ax ch ax dx=\dfrac{sh^{2} ax}{2a} +C $ | |
| $ \int sh px ch qx dx=\dfrac{ch(p+q)x}{2(p+q)}+\dfrac{ch(p-q)x}{2(p-q)} +C $ | |
| $ \int sh^{n} ax ch ax dx=\dfrac{sh^{n+1} ax}{(n+1)a} +C $ | |
| $ \int ch^{n} ax sh ax dx=\dfrac{ch^{n+1} ax}{(n+1)a} +C $ | |
| $ \int sh^{2} ax ch^{2} ax dx=\dfrac{sh4ax}{32a}-\dfrac{x}{8} +C $ | |
| $ \int\dfrac{dx}{sh ax ch ax}=\dfrac{1}{a}\ln th ax +C $ | |
| $ \int\dfrac{dx}{sh^{2} ax ch ax}=-\dfrac{1}{a}Arc tg sh ax-\dfrac{1}{a sh ax} +C $ | |
| $ \int\dfrac{dx}{sh ax ch^{2} ax}=\dfrac{1}{a ch ax}+\dfrac{1}{a}\ln th\dfrac{ax}{2} +C $ | |
| $ \int\dfrac{dx}{sh^{2} ax ch^{2} ax}=-\dfrac{2 coth2ax}{a} +C $ | |
| $ \int\dfrac{sh^{2} ax dx}{ch ax}=-\dfrac{1}{a}Arc tg sh ax+\dfrac{sh ax}{a} +C $ | |
| $ \int\dfrac{ch^{2} ax dx}{sh ax}=\dfrac{1}{a}\ln th\dfrac{ax}{2}+\dfrac{ch ax}{a} +C $ | |
| $ \int\dfrac{dx}{sh ax(ch ax+1)}=\dfrac{1}{2a}\ln th\dfrac{ax}{2}+\dfrac{1}{2a(ch ax+1)} +C $ | |
| $ \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} +C $ | |
| $ \int\dfrac{dx}{sh ax(ch ax-1)}=-\dfrac{1}{2a}\ln th\dfrac{ax}{2}-\dfrac{1}{2a(ch ax-1)} +C $ | |
| Inverse Hyperbolic Sine ( arg sh x) | |
|---|---|
| $ \int\arg sh\dfrac{x}{a}dx=x\arg sh\dfrac{x}{a}-\sqrt{x^{2}+a^{2}} +C $ | |
| $ \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} +C $ | |
| $ \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} +C $ | |
| $ \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} +C $ | |
| $ \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) +C $ | |
