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| <math> \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} </math> | | <math> \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} </math> | ||
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| + | | <math> \int sh ax ch ax dx=\dfrac{sh^{2} ax}{2a} +C</math> | ||
| + | |- | ||
| + | | <math> \int sh px ch qx dx=\dfrac{ch(p+q)x}{2(p+q)}+\dfrac{ch(p-q)x}{2(p-q)} +C</math> | ||
| + | |- | ||
| + | | <math> \int sh^{n} ax ch ax dx=\dfrac{sh^{n+1} ax}{(n+1)a} +C</math> | ||
| + | |- | ||
| + | | <math> \int ch^{n} ax sh ax dx=\dfrac{ch^{n+1} ax}{(n+1)a} +C</math> | ||
| + | |- | ||
| + | | <math> \int sh^{2} ax ch^{2} ax dx=\dfrac{sh4ax}{32a}-\dfrac{x}{8} +C</math> | ||
| + | |- | ||
| + | | <math> \int\dfrac{dx}{sh ax ch ax}=\dfrac{1}{a}\ln th ax +C</math> | ||
| + | |- | ||
| + | | <math> \int\dfrac{dx}{sh^{2} ax ch ax}=-\dfrac{1}{a}Arc tg sh ax-\dfrac{1}{a sh ax} +C</math> | ||
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| + | | <math> \int\dfrac{dx}{sh ax ch^{2} ax}=\dfrac{1}{a ch ax}+\dfrac{1}{a}\ln th\dfrac{ax}{2} +C</math> | ||
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| + | | <math> \int\dfrac{dx}{sh^{2} ax ch^{2} ax}=-\dfrac{2 coth2ax}{a} +C</math> | ||
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| + | | <math> \int\dfrac{sh^{2} ax dx}{ch ax}=-\dfrac{1}{a}Arc tg sh ax+\dfrac{sh ax}{a} +C</math> | ||
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| + | | <math> \int\dfrac{ch^{2} ax dx}{sh ax}=\dfrac{1}{a}\ln th\dfrac{ax}{2}+\dfrac{ch ax}{a} +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> | ||
| + | |- | ||
| + | | <math> \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</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> | ||
| + | |- | ||
| + | ! 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 Cosine ( arg ch x) | ||
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| + | | <math> \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} +C</math> | ||
| + | |- | ||
| + | | <math> \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} +C</math> | ||
| + | |- | ||
| + | | <math> \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} +C</math> | ||
| + | |- | ||
| + | | <math> \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] +C</math> | ||
| + | |- | ||
| + | | <math> \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) +C</math> | ||
|} | |} | ||
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Latest revision as of 16:44, 26 February 2015
Indefinite Integrals with hyperbolic cosine (ch x)
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| $ \int ch ax dx=\dfrac{sh ax}{a} +C $ | |
| $ \int x ch ax dx=\dfrac{x sh ax}{a}-\dfrac{ch ax}{a^{2}} +C $ | |
| $ \int x^{2} ch ax dx=(\dfrac{x^{2}}{a^{2}}+\dfrac{2}{a^{3}}) sh ax-\dfrac{2x}{a^{2}} ch ax +C $ | |
| $ \int\dfrac{ch ax}{x} dx=\ln x+\dfrac{(ax)^{2}}{2\cdot2!}+\dfrac{(ax)^{4}}{4\cdot4!}+\cdots +C $ | |
| $ \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} +C $ | |
| $ \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)!}\} +C $ | |
| $ \int ch^{2} ax dx=\dfrac{sh ax ch ax}{2a}+\dfrac{x}{2} +C $ | |
| $ \int x ch^{2} ax dx=\dfrac{x sh2ax}{4a}-\dfrac{ch2ax}{8a^{2}}+\dfrac{x^{2}}{4} +C $ | |
| $ \int\dfrac{dx}{ch^{2} ax}=\dfrac{th ax}{a} +C $ | |
| $ \int ch ax ch px dx=\dfrac{sh(a-p) x}{2(a-p)}-\dfrac{sh(a+p)x}{2(a+p)} +C $ | |
| $ \int ch ax sin px dx=\dfrac{a ch ax sin px-p sh ax cos px}{a^{2}+p^{2}} +C $ | |
| $ \int ch ax cos px dx=\dfrac{a sh ax cos px+p ch ax sin px}{a^{2}+p^{2}} +C $ | |
| $ \int\dfrac{dx}{ch ax+1}=\dfrac{1}{a} th\dfrac{ax}{2} +C $ | |
| $ \int\dfrac{dx}{(ch ax-1)}=-\dfrac{1}{a} coth\dfrac{ax}{2} +C $ | |
| $ \int\dfrac{xdx}{(ch ax+1)}=\dfrac{x}{a} th\dfrac{ax}{2}-\dfrac{2}{a^{2}}\ln ch\dfrac{ax}{2} +C $ | |
| $ \int\dfrac{xdx}{(ch ax-1)}=-\dfrac{x}{a}coth\dfrac{ax}{2}+\dfrac{2}{a^{2}}\ln sh\dfrac{ax}{2} +C $ | |
| $ \int\dfrac{dx}{(ch ax+1)^{2}}=\dfrac{1}{2a}th\dfrac{ax}{2}-\dfrac{1}{6a}th^{3}\dfrac{ax}{2} +C $ | |
| $ \int\dfrac{dx}{(ch ax-1)^{2}}=\dfrac{1}{2a}coth\dfrac{ax}{2}-\dfrac{1}{6a}coth^{3}\dfrac{ax}{2} +C $ | |
| $ \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 +C $ | |
| $ \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} $ | |
| $ \int\dfrac{1}{ch ax}dx=\dfrac{2}{a}Arc tg e^{ax} +C $ | |
| $ \int\dfrac{1}{ch^{2} ax}dx=\dfrac{th ax}{a} +C $ | |
| $ \int\dfrac{1}{ch^{3} ax}dx=\dfrac{th ax}{2a ch ax}+\dfrac{1}{2a}Arc tg sh ax+C $ | |
| $ \int\dfrac{th ax}{ch^{n} ax}dx=-\dfrac{1}{na ch^{n} ax} +C $ | |
| $ \int ch ax dx=\dfrac{sh ax}{a} +C $ | |
| $ \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\} +C $ | |
| $ \int\dfrac{xdx}{ch^{2} ax}=\dfrac{x th ax}{a}-\dfrac{1}{a^{2}}\ln ch ax +C $ | |
| $ \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\} +C $ | |
| $ \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} $ | |
| $ \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 Cosine ( arg ch x) | |
|---|---|
| $ \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} +C $ | |
| $ \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} +C $ | |
| $ \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} +C $ | |
| $ \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] +C $ | |
| $ \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) +C $ | |
