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Revision as of 17:17, 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} $ |
