Seznam integrálů hyperbolických funkcí

Seznamy integrálů
Tabulka integrálů elementárních funkcí
racionální funkce iracionální funkce exponenciální funkce logaritmické funkce trigonometrické funkce inverzní trigonometrické funkce hyperbolické funkce inverzní hyperbolické funkce

Toto je seznam integrálů (primitivních funkcí) hyperbolických funkcí.

$\int \sinh cx\,\mathrm {d} x={\frac {1}{c}}\cosh cx$$\int \sinh cx\,\mathrm {d} x={\frac {1}{c}}\cosh cx$

$\int \cosh cx\,\mathrm {d} x={\frac {1}{c}}\sinh cx$$\int \cosh cx\,\mathrm {d} x={\frac {1}{c}}\sinh cx$

$\int \sinh ^{2}cx\,\mathrm {d} x={\frac {1}{4c}}\sinh 2cx-{\frac {x}{2}}$$\int \sinh ^{2}cx\,\mathrm {d} x={\frac {1}{4c}}\sinh 2cx-{\frac {x}{2}}$

$\int \cosh ^{2}cx\,\mathrm {d} x={\frac {1}{4c}}\sinh 2cx+{\frac {x}{2}}$$\int \cosh ^{2}cx\,\mathrm {d} x={\frac {1}{4c}}\sinh 2cx+{\frac {x}{2}}$

$\int \sinh ^{n}cx\,\mathrm {d} x={\frac {1}{cn}}\sinh ^{n-1}cx\cosh cx-{\frac {n-1}{n}}\int \sinh ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(pro }}n>0{\mbox{)}}$$\int \sinh ^{n}cx\,\mathrm {d} x={\frac {1}{cn}}\sinh ^{n-1}cx\cosh cx-{\frac {n-1}{n}}\int \sinh ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(pro }}n>0{\mbox{)}}$

také: $\int \sinh ^{n}cx\,\mathrm {d} x={\frac {1}{c(n+1)}}\sinh ^{n+1}cx\cosh cx-{\frac {n+2}{n+1}}\int \sinh ^{n+2}cx\,\mathrm {d} x\qquad {\mbox{(pro }}n<0{\mbox{, }}n\neq -1{\mbox{)}}$$\int \sinh ^{n}cx\,\mathrm {d} x={\frac {1}{c(n+1)}}\sinh ^{n+1}cx\cosh cx-{\frac {n+2}{n+1}}\int \sinh ^{n+2}cx\,\mathrm {d} x\qquad {\mbox{(pro }}n<0{\mbox{, }}n\neq -1{\mbox{)}}$

$\int \cosh ^{n}cx\,\mathrm {d} x={\frac {1}{cn}}\sinh cx\cosh ^{n-1}cx+{\frac {n-1}{n}}\int \cosh ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(pro }}n>0{\mbox{)}}$$\int \cosh ^{n}cx\,\mathrm {d} x={\frac {1}{cn}}\sinh cx\cosh ^{n-1}cx+{\frac {n-1}{n}}\int \cosh ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(pro }}n>0{\mbox{)}}$

také: $\int \cosh ^{n}cx\,\mathrm {d} x=-{\frac {1}{c(n+1)}}\sinh cx\cosh ^{n+1}cx-{\frac {n+2}{n+1}}\int \cosh ^{n+2}cx\,\mathrm {d} x\qquad {\mbox{(pro }}n<0{\mbox{, }}n\neq -1{\mbox{)}}$$\int \cosh ^{n}cx\,\mathrm {d} x=-{\frac {1}{c(n+1)}}\sinh cx\cosh ^{n+1}cx-{\frac {n+2}{n+1}}\int \cosh ^{n+2}cx\,\mathrm {d} x\qquad {\mbox{(pro }}n<0{\mbox{, }}n\neq -1{\mbox{)}}$

$\int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\ln \left|\tanh {\frac {cx}{2}}\right|$$\int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\ln \left|\tanh {\frac {cx}{2}}\right|$

také: $\int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\sinh cx}}\right|$$\int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\sinh cx}}\right|$

také: $\int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\sinh cx}{\cosh cx+1}}\right|$$\int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\sinh cx}{\cosh cx+1}}\right|$

také: $\int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\cosh cx+1}}\right|$$\int {\frac {\mathrm {d} x}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\cosh cx+1}}\right|$

$\int {\frac {\mathrm {d} x}{\cosh cx}}={\frac {2}{c}}\arctan e^{cx}$$\int {\frac {\mathrm {d} x}{\cosh cx}}={\frac {2}{c}}\arctan e^{cx}$

$\int {\frac {\mathrm {d} x}{\sinh ^{n}cx}}={\frac {\cosh cx}{c(n-1)\sinh ^{n-1}cx}}-{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\sinh ^{n-2}cx}}\qquad {\mbox{(pro }}n\neq 1{\mbox{)}}$$\int {\frac {\mathrm {d} x}{\sinh ^{n}cx}}={\frac {\cosh cx}{c(n-1)\sinh ^{n-1}cx}}-{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\sinh ^{n-2}cx}}\qquad {\mbox{(pro }}n\neq 1{\mbox{)}}$

$\int {\frac {\mathrm {d} x}{\cosh ^{n}cx}}={\frac {\sinh cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\cosh ^{n-2}cx}}\qquad {\mbox{(pro }}n\neq 1{\mbox{)}}$$\int {\frac {\mathrm {d} x}{\cosh ^{n}cx}}={\frac {\sinh cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\cosh ^{n-2}cx}}\qquad {\mbox{(pro }}n\neq 1{\mbox{)}}$

$\int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}\mathrm {d} x={\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}}\mathrm {d} x\qquad {\mbox{(pro }}m\neq n{\mbox{)}}$$\int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}\mathrm {d} x={\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}}\mathrm {d} x\qquad {\mbox{(pro }}m\neq n{\mbox{)}}$

také: $\int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}\mathrm {d} x=-{\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}}\mathrm {d} x\qquad {\mbox{(pro }}m\neq 1{\mbox{)}}$$\int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}\mathrm {d} x=-{\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}}\mathrm {d} x\qquad {\mbox{(pro }}m\neq 1{\mbox{)}}$

také: $\int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}\mathrm {d} x=-{\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}}\mathrm {d} x\qquad {\mbox{(pro }}m\neq 1{\mbox{)}}$$\int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}\mathrm {d} x=-{\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}}\mathrm {d} x\qquad {\mbox{(pro }}m\neq 1{\mbox{)}}$

$\int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}\mathrm {d} x={\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}}\mathrm {d} x\qquad {\mbox{(pro }}m\neq n{\mbox{)}}$$\int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}\mathrm {d} x={\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}}\mathrm {d} x\qquad {\mbox{(pro }}m\neq n{\mbox{)}}$

také: $\int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}\mathrm {d} x={\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}}\mathrm {d} x\qquad {\mbox{(pro }}n\neq 1{\mbox{)}}$$\int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}\mathrm {d} x={\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}}\mathrm {d} x\qquad {\mbox{(pro }}n\neq 1{\mbox{)}}$

také: $\int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}\mathrm {d} x=-{\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}}\mathrm {d} x\qquad {\mbox{(pro }}n\neq 1{\mbox{)}}$$\int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}\mathrm {d} x=-{\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}}\mathrm {d} x\qquad {\mbox{(pro }}n\neq 1{\mbox{)}}$

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

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

$\int \tanh cx\,\mathrm {d} x={\frac {1}{c}}\ln |\cosh cx|$$\int \tanh cx\,\mathrm {d} x={\frac {1}{c}}\ln |\cosh cx|$

$\int \coth cx\,\mathrm {d} x={\frac {1}{c}}\ln |\sinh cx|$$\int \coth cx\,\mathrm {d} x={\frac {1}{c}}\ln |\sinh cx|$

$\int \tanh ^{n}cx\,\mathrm {d} x=-{\frac {1}{c(n-1)}}\tanh ^{n-1}cx+\int \tanh ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(pro }}n\neq 1{\mbox{)}}$$\int \tanh ^{n}cx\,\mathrm {d} x=-{\frac {1}{c(n-1)}}\tanh ^{n-1}cx+\int \tanh ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(pro }}n\neq 1{\mbox{)}}$

$\int \coth ^{n}cx\,\mathrm {d} x=-{\frac {1}{c(n-1)}}\coth ^{n-1}cx+\int \coth ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(pro }}n\neq 1{\mbox{)}}$$\int \coth ^{n}cx\,\mathrm {d} x=-{\frac {1}{c(n-1)}}\coth ^{n-1}cx+\int \coth ^{n-2}cx\,\mathrm {d} x\qquad {\mbox{(pro }}n\neq 1{\mbox{)}}$

$\int \sinh bx\sinh cx\,\mathrm {d} x={\frac {1}{b^{2}-c^{2}}}(b\sinh cx\cosh bx-c\cosh cx\sinh bx)\qquad {\mbox{(pro }}b^{2}\neq c^{2}{\mbox{)}}$$\int \sinh bx\sinh cx\,\mathrm {d} x={\frac {1}{b^{2}-c^{2}}}(b\sinh cx\cosh bx-c\cosh cx\sinh bx)\qquad {\mbox{(pro }}b^{2}\neq c^{2}{\mbox{)}}$

$\int \cosh bx\cosh cx\,\mathrm {d} x={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\cosh cx-c\sinh cx\cosh bx)\qquad {\mbox{(pro }}b^{2}\neq c^{2}{\mbox{)}}$$\int \cosh bx\cosh cx\,\mathrm {d} x={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\cosh cx-c\sinh cx\cosh bx)\qquad {\mbox{(pro }}b^{2}\neq c^{2}{\mbox{)}}$

$\int \cosh bx\sinh cx\,\mathrm {d} x={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\sinh cx-c\cosh bx\cosh cx)\qquad {\mbox{(pro }}b^{2}\neq c^{2}{\mbox{)}}$$\int \cosh bx\sinh cx\,\mathrm {d} x={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\sinh cx-c\cosh bx\cosh cx)\qquad {\mbox{(pro }}b^{2}\neq c^{2}{\mbox{)}}$

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