Giperbolik funksiyalarning integrallar jadvali

Izoh. Hamma integrallarda oʻzgarmas qoʻshiluvchi tushirib qoldirilgan.

$\int \mathrm{sh}cxdx=\frac{1}{c}\mathrm{ch}cx$

$\int \mathrm{ch}cxdx=\frac{1}{c}\mathrm{sh}cx$

$\int {\mathrm{sh}}^{2}cxdx=\frac{1}{4c}\mathrm{sh}2cx-\frac{x}{2}$

$\int {\mathrm{ch}}^{2}cxdx=\frac{1}{4c}\mathrm{sh}2cx+\frac{x}{2}$

$\int {\mathrm{sh}}^{n}cxdx=\frac{1}{cn}{\mathrm{sh}}^{n-1}cx\mathrm{ch}cx-\frac{n-1}{n}\int {\mathrm{sh}}^{n-2}cxdx\text{( }n>0\text{)}$

также: $\int {\mathrm{sh}}^{n}cxdx=\frac{1}{c(n+1)}{\mathrm{sh}}^{n+1}cx\mathrm{ch}cx-\frac{n+2}{n+1}\int {\mathrm{sh}}^{n+2}cxdx\text{( }n<0\text{, }n\ne -1\text{)}$

$\int {\mathrm{ch}}^{n}cxdx=\frac{1}{cn}\mathrm{sh}cx{\mathrm{ch}}^{n-1}cx+\frac{n-1}{n}\int {\mathrm{ch}}^{n-2}cxdx\text{( }n>0\text{)}$

также: $\int {\mathrm{ch}}^{n}cxdx=-\frac{1}{c(n+1)}\mathrm{sh}cx{\mathrm{ch}}^{n+1}cx-\frac{n+2}{n+1}\int {\mathrm{ch}}^{n+2}cxdx\text{(}n<0\text{, }n\ne -1\text{)}$

$\int \frac{dx}{\mathrm{sh}cx}=\frac{1}{c}\ln |\mathrm{th}\frac{cx}{2}|=\frac{1}{c}\ln \left|\frac{\mathrm{ch}cx-1}{\mathrm{sh}cx}\right|=\frac{1}{c}\ln \left|\frac{\mathrm{sh}cx}{\mathrm{ch}cx+1}\right|=\frac{1}{c}\ln \left|\frac{\mathrm{ch}cx-1}{\mathrm{ch}cx+1}\right|$

$\int \frac{dx}{{\mathrm{sh}}^{2}cx}=-\frac{1}{c}\mathrm{cth}cx$

$\int \frac{dx}{\mathrm{ch}cx}=\frac{2}{c}\mathrm{arctg}{e}^{cx}$

$\int \frac{dx}{{\mathrm{ch}}^{2}cx}=\frac{1}{c}\mathrm{th}cx$

$\int \frac{dx}{{\mathrm{sh}}^{n}cx}=\frac{\mathrm{ch}cx}{c(n-1){\mathrm{sh}}^{n-1}cx}-\frac{n-2}{n-1}\int \frac{dx}{{\mathrm{sh}}^{n-2}cx}\text{( }n\ne 1\text{)}$

$\int \frac{dx}{{\mathrm{ch}}^{n}cx}=\frac{\mathrm{sh}cx}{c(n-1){\mathrm{ch}}^{n-1}cx}+\frac{n-2}{n-1}\int \frac{dx}{{\mathrm{ch}}^{n-2}cx}\text{( }n\ne 1\text{)}$

$\int \frac{{\mathrm{ch}}^{n}cx}{{\mathrm{sh}}^{m}cx}dx=\frac{{\mathrm{ch}}^{n-1}cx}{c(n-m){\mathrm{sh}}^{m-1}cx}+\frac{n-1}{n-m}\int \frac{{\mathrm{ch}}^{n-2}cx}{{\mathrm{sh}}^{m}cx}dx\text{( }m\ne n\text{)}$

также: $\int \frac{{\mathrm{ch}}^{n}cx}{{\mathrm{sh}}^{m}cx}dx=-\frac{{\mathrm{ch}}^{n+1}cx}{c(m-1){\mathrm{sh}}^{m-1}cx}+\frac{n-m+2}{m-1}\int \frac{{\mathrm{ch}}^{n}cx}{{\mathrm{sh}}^{m-2}cx}dx\text{( }m\ne 1\text{)}$

также: $\int \frac{{\mathrm{ch}}^{n}cx}{{\mathrm{sh}}^{m}cx}dx=-\frac{{\mathrm{ch}}^{n-1}cx}{c(m-1){\mathrm{sh}}^{m-1}cx}+\frac{n-1}{m-1}\int \frac{{\mathrm{ch}}^{n-2}cx}{{\mathrm{sh}}^{m-2}cx}dx\text{( }m\ne 1\text{)}$

$\int \frac{{\mathrm{sh}}^{m}cx}{{\mathrm{ch}}^{n}cx}dx=\frac{{\mathrm{sh}}^{m-1}cx}{c(m-n){\mathrm{ch}}^{n-1}cx}+\frac{m-1}{m-n}\int \frac{{\mathrm{sh}}^{m-2}cx}{{\mathrm{ch}}^{n}cx}dx\text{( }m\ne n\text{)}$

также: $\int \frac{{\mathrm{sh}}^{m}cx}{{\mathrm{ch}}^{n}cx}dx=\frac{{\mathrm{sh}}^{m+1}cx}{c(n-1){\mathrm{ch}}^{n-1}cx}+\frac{m-n+2}{n-1}\int \frac{{\mathrm{sh}}^{m}cx}{{\mathrm{ch}}^{n-2}cx}dx\text{( }n\ne 1\text{)}$

также: $\int \frac{{\mathrm{sh}}^{m}cx}{{\mathrm{ch}}^{n}cx}dx=-\frac{{\mathrm{sh}}^{m-1}cx}{c(n-1){\mathrm{ch}}^{n-1}cx}+\frac{m-1}{n-1}\int \frac{{\mathrm{sh}}^{m-2}cx}{{\mathrm{ch}}^{n-2}cx}dx\text{( }n\ne 1\text{)}$

$\int x\mathrm{sh}cxdx=\frac{1}{c}x\mathrm{ch}cx-\frac{1}{c^2}\mathrm{sh}cx$

$\int x\mathrm{ch}cxdx=\frac{1}{c}x\mathrm{sh}cx-\frac{1}{c^2}\mathrm{ch}cx$

$\int \mathrm{th}cxdx=\frac{1}{c}\ln |\mathrm{ch}cx|$

$\int \mathrm{cth}cxdx=\frac{1}{c}\ln |\mathrm{sh}cx|$

$\int {\mathrm{th}}^{2}cxdx=x-\frac{1}{c}\mathrm{th}cx$

$\int {\mathrm{cth}}^{2}cxdx=x-\frac{1}{c}\mathrm{cth}cx$

$\int {\mathrm{th}}^{n}cxdx=-\frac{1}{c(n-1)}{\mathrm{th}}^{n-1}cx+\int {\mathrm{th}}^{n-2}cxdx\text{( }n\ne 1\text{ )}$

$\int {\mathrm{cth}}^{n}cxdx=-\frac{1}{c(n-1)}{\mathrm{cth}}^{n-1}cx+\int {\mathrm{cth}}^{n-2}cxdx\text{( }n\ne 1\text{)}$

$\int \mathrm{sh}bx\mathrm{sh}cxdx=\frac{1}{b^2-c^2}(b\mathrm{sh}cx\mathrm{ch}bx-c\mathrm{ch}cx\mathrm{sh}bx)\text{( }{b}^2\ne c^2\text{)}$

$\int \mathrm{ch}bx\mathrm{ch}cxdx=\frac{1}{b^2-c^2}(b\mathrm{sh}bx\mathrm{ch}cx-c\mathrm{sh}cx\mathrm{ch}bx)\text{( }{b}^2\ne c^2\text{)}$

$\int \mathrm{ch}bx\mathrm{sh}cxdx=\frac{1}{b^2-c^2}(b\mathrm{sh}bx\mathrm{sh}cx-c\mathrm{ch}bx\mathrm{ch}cx)\text{( }{b}^2\ne c^2\text{)}$

$\int \mathrm{sh}(ax+b)\sin (cx+d)dx=\frac{a}{a^2+c^2}\mathrm{ch}(ax+b)\sin (cx+d)-\frac{c}{a^2+c^2}\mathrm{sh}(ax+b)\cos (cx+d)$

$\int \mathrm{sh}(ax+b)\cos (cx+d)dx=\frac{a}{a^2+c^2}\mathrm{ch}(ax+b)\cos (cx+d)+\frac{c}{a^2+c^2}\mathrm{sh}(ax+b)\sin (cx+d)$

$\int \mathrm{ch}(ax+b)\sin (cx+d)dx=\frac{a}{a^2+c^2}\mathrm{sh}(ax+b)\sin (cx+d)-\frac{c}{a^2+c^2}\mathrm{ch}(ax+b)\cos (cx+d)$

$\int \mathrm{ch}(ax+b)\cos (cx+d)dx=\frac{a}{a^2+c^2}\mathrm{sh}(ax+b)\cos (cx+d)+\frac{c}{a^2+c^2}\mathrm{ch}(ax+b)\sin (cx+d)$