Formelsammlung Mathematik: Unbestimmte Integrale hyperbolischer Funktionen

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Nachfolgende Liste enthält einige Integrale hyperbolischer Funktionen.

$\int \sinh cxdx=\frac{1}{c}\cosh cx$

$\int \cosh cxdx=\frac{1}{c}\sinh cx$

$\int {\sinh }^{2}cxdx=\frac{1}{2c}\sinh cx\cosh cx-\frac{x}{2}$

$\int {\cosh }^{2}cxdx=\frac{1}{2c}\sinh cx\cosh cx+\frac{x}{2}$

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

oder: $\int {\sinh }^{n}cxdx=\frac{1}{c(n+1)}{\sinh }^{n+1}cx\cosh cx-\frac{n+2}{n+1}\int {\sinh }^{n+2}cxdx\text{( }n<0\text{, }n\ne -1\text{)}$

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

oder: $\int {\cosh }^{n}cxdx=-\frac{1}{c(n+1)}\sinh cx{\cosh }^{n+1}cx+\frac{n+2}{n+1}\int {\cosh }^{n+2}cxdx\text{( }n<0\text{, }n\ne -1\text{)}$

$\int \frac{dx}{\sinh cx}=\frac{1}{c}\ln |\tanh \frac{cx}{2}|$

oder: $\int \frac{dx}{\sinh cx}=\frac{1}{c}\ln \left|\frac{\cosh cx-1}{\sinh cx}\right|$

oder: $\int \frac{dx}{\sinh cx}=\frac{1}{c}\ln \left|\frac{\sinh cx}{\cosh cx+1}\right|$

oder: $\int \frac{dx}{\sinh cx}=\frac{1}{c}\ln \left|\frac{\cosh cx-1}{\cosh cx+1}\right|$

$\int \frac{dx}{\cosh cx}=\frac{2}{c}\arctan e^{cx}$

$\int \frac{dx}{{\sinh }^{n}cx}=\frac{\cosh cx}{c(n-1){\sinh }^{n-1}cx}-\frac{n-2}{n-1}\int \frac{dx}{{\sinh }^{n-2}cx}\text{( }n\ne 1\text{)}$

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

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

oder: $\int \frac{{\cosh }^{n}cx}{{\sinh }^{m}cx}dx=-\frac{{\cosh }^{n+1}cx}{c(m-1){\sinh }^{m-1}cx}+\frac{n-m+2}{m-1}\int \frac{{\cosh }^{n}cx}{{\sinh }^{m-2}cx}dx\text{( }m\ne 1\text{)}$

oder: $\int \frac{{\cosh }^{n}cx}{{\sinh }^{m}cx}dx=-\frac{{\cosh }^{n-1}cx}{c(m-1){\sinh }^{m-1}cx}+\frac{n-1}{m-1}\int \frac{{\cosh }^{n-2}cx}{{\sinh }^{m-2}cx}dx\text{( }m\ne 1\text{)}$

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

oder: $\int \frac{{\sinh }^{m}cx}{{\cosh }^{n}cx}dx=\frac{{\sinh }^{m+1}cx}{c(n-1){\cosh }^{n-1}cx}+\frac{m-n+2}{n-1}\int \frac{{\sinh }^{m}cx}{{\cosh }^{n-2}cx}dx\text{( }n\ne 1\text{)}$

oder: $\int \frac{{\sinh }^{m}cx}{{\cosh }^{n}cx}dx=-\frac{{\sinh }^{m-1}cx}{c(n-1){\cosh }^{n-1}cx}+\frac{m-1}{n-1}\int \frac{{\sinh }^{m-2}cx}{{\cosh }^{n-2}cx}dx\text{( }n\ne 1\text{)}$

$\int x\sinh cxdx=\frac{1}{c}x\cosh cx-\frac{1}{c^2}\sinh cx$

$\int x\cosh cxdx=\frac{1}{c}x\sinh cx-\frac{1}{c^2}\cosh cx$

$\int \tanh cxdx=\frac{1}{c}\ln |\cosh cx|$

$\int \coth cxdx=\frac{1}{c}\ln |\sinh cx|$

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

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

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

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

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

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

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

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

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

Siehe auch

• Englische Wikipedia