Lijst van integralen van hyperbolische functies

Dit artikel bevat een lijst van integralen van hyperbolische functies.

${\displaystyle \int \sinh ax\,\mathrm {d} x={\frac {1}{a}}\cosh ax+C}$

${\displaystyle \int \cosh ax\,\mathrm {d} x={\frac {1}{a}}\sinh ax+C}$

${\displaystyle \int \sinh ^{2}ax\,\mathrm {d} x={\frac {1}{4a}}\sinh 2ax-{\frac {x}{2}}+C}$

${\displaystyle \int \cosh ^{2}ax\,\mathrm {d} x={\frac {1}{4a}}\sinh 2ax+{\frac {x}{2}}+C}$

${\displaystyle \int \tanh ^{2}ax\,\mathrm {d} x=x-{\frac {\tanh ax}{a}}+C}$

${\displaystyle \int \sinh ^{n}ax\,\mathrm {d} x={\frac {1}{an}}\sinh ^{n-1}ax\cosh ax-{\frac {n-1}{n}}\int \sinh ^{n-2}ax\,\mathrm {d} x\qquad {\mbox{(voor }}n>0{\mbox{)}}}$

${\displaystyle \int \sinh ^{n}ax\,\mathrm {d} x={\frac {1}{a(n+1)}}\sinh ^{n+1}ax\cosh ax-{\frac {n+2}{n+1}}\int \sinh ^{n+2}ax\,\mathrm {d} x\qquad {\mbox{(voor }}n<0{\mbox{, }}n\neq -1{\mbox{)}}}$

${\displaystyle \int \cosh ^{n}ax\,\mathrm {d} x={\frac {1}{an}}\sinh ax\cosh ^{n-1}ax+{\frac {n-1}{n}}\int \cosh ^{n-2}ax\,\mathrm {d} x\qquad {\mbox{(voor }}n>0{\mbox{)}}}$

${\displaystyle \int \cosh ^{n}ax\,\mathrm {d} x=-{\frac {1}{a(n+1)}}\sinh ax\cosh ^{n+1}ax-{\frac {n+2}{n+1}}\int \cosh ^{n+2}ax\,\mathrm {d} x\qquad {\mbox{(voor }}n<0{\mbox{, }}n\neq -1{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2}}\right|+C}$

${\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\sinh ax}}\right|+C}$

${\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\sinh ax}{\cosh ax+1}}\right|+C}$

${\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\cosh ax+1}}\right|+C}$

${\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {2}{a}}\arctan e^{ax}+C}$

${\displaystyle \int {\frac {dx}{\sinh ^{n}ax}}=-{\frac {\cosh ax}{a(n-1)\sinh ^{n-1}ax}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}ax}}\qquad {\mbox{(voor }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {dx}{\cosh ^{n}ax}}={\frac {\sinh ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}ax}}\qquad {\mbox{(voor }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}\mathrm {d} x={\frac {\cosh ^{n-1}ax}{a(n-m)\sinh ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m}ax}}\mathrm {d} x\qquad {\mbox{(voor }}m\neq n{\mbox{)}}}$

${\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}\mathrm {d} x=-{\frac {\cosh ^{n+1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}ax}{\sinh ^{m-2}ax}}\mathrm {d} x\qquad {\mbox{(voor }}m\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}\mathrm {d} x=-{\frac {\cosh ^{n-1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m-2}ax}}\mathrm {d} x\qquad {\mbox{(voor }}m\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}\mathrm {d} x={\frac {\sinh ^{m-1}ax}{a(m-n)\cosh ^{n-1}ax}}+{\frac {m-1}{n-m}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n}ax}}\mathrm {d} x\qquad {\mbox{(voor }}m\neq n{\mbox{)}}}$

${\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}\mathrm {d} x={\frac {\sinh ^{m+1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}ax}{\cosh ^{n-2}ax}}\mathrm {d} x\qquad {\mbox{(voor }}n\neq 1{\mbox{)}}}$

${\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}\mathrm {d} x=-{\frac {\sinh ^{m-1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n-2}ax}}\mathrm {d} x\qquad {\mbox{(voor }}n\neq 1{\mbox{)}}}$

${\displaystyle \int x\sinh ax\,\mathrm {d} x={\frac {1}{a}}x\cosh ax-{\frac {1}{a^{2}}}\sinh ax+C}$

${\displaystyle \int x\cosh ax\,\mathrm {d} x={\frac {1}{a}}x\sinh ax-{\frac {1}{a^{2}}}\cosh ax+C}$

${\displaystyle \int x^{2}\cosh ax\,\mathrm {d} x=-{\frac {2x\cosh ax}{a^{2}}}+\left({\frac {x^{2}}{a}}+{\frac {2}{a^{3}}}\right)\sinh ax+C}$

${\displaystyle \int \tanh ax\,\mathrm {d} x={\frac {1}{a}}\ln |\cosh ax|+C}$

${\displaystyle \int \coth ax\,\mathrm {d} x={\frac {1}{a}}\ln |\sinh ax|+C}$

${\displaystyle \int \tanh ^{n}ax\,\mathrm {d} x=-{\frac {1}{a(n-1)}}\tanh ^{n-1}ax+\int \tanh ^{n-2}ax\,\mathrm {d} x\qquad {\mbox{(voor }}n\neq 1{\mbox{)}}}$

${\displaystyle \int \coth ^{n}ax\,\mathrm {d} x=-{\frac {1}{a(n-1)}}\coth ^{n-1}ax+\int \coth ^{n-2}ax\,\mathrm {d} x\qquad {\mbox{(voor }}n\neq 1{\mbox{)}}}$

${\displaystyle \int \sinh ax\sinh bx\,\mathrm {d} x={\frac {1}{a^{2}-b^{2}}}(a\sinh bx\cosh ax-b\cosh bx\sinh ax)+C\qquad {\mbox{(voor }}a^{2}\neq b^{2}{\mbox{)}}}$

${\displaystyle \int \cosh ax\cosh bx\,\mathrm {d} x={\frac {1}{a^{2}-b^{2}}}(a\sinh ax\cosh bx-b\sinh bx\cosh ax)+C\qquad {\mbox{(voor }}a^{2}\neq b^{2}{\mbox{)}}}$

${\displaystyle \int \cosh ax\sinh bx\,\mathrm {d} x={\frac {1}{a^{2}-b^{2}}}(a\sinh ax\sinh bx-b\cosh ax\cosh bx)+C\qquad {\mbox{(voor }}a^{2}\neq b^{2}{\mbox{)}}}$

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

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

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

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