Daftar integral dari fungsi hiperbolik

Daftar integral (antiderivatif) dari fungsi hiperbolik. Untuk daftar lengkap fungsi integral, lihat Tabel integral.

Dalam semua rumus, konstanta a diasumsikan bukan nol, dan C melambangkan konstanta integrasi.

Integral melibatkan hanya fungsi hiperbolik sinus

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

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

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

juga: ${\displaystyle \int \sinh ^{n}ax\,dx={\frac {1}{a(n+1)}}\sinh ^{n+1}ax\cosh ax-{\frac {n+2}{n+1}}\int \sinh ^{n+2}ax\,dx\qquad {\mbox{(for }}n<0{\mbox{, }}n\neq -1{\mbox{)}}\,}$

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

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

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

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

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

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

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

Integral melibatkan hanya fungsi hiperbolik kosinus

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

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

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

juga: ${\displaystyle \int \cosh ^{n}ax\,dx=-{\frac {1}{a(n+1)}}\sinh ax\cosh ^{n+1}ax+{\frac {n+2}{n+1}}\int \cosh ^{n+2}ax\,dx\qquad {\mbox{(for }}n<0{\mbox{, }}n\neq -1{\mbox{)}}\,}$

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

juga: ${\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {1}{a}}\arctan(\sinh ax)+C\,}$

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

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

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

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

Integral lain-lain

Integral fungsi hiperbolik tangen, kotangen, sekan, kosekan

${\displaystyle \int \tanh x\,dx=\ln \cosh x+C}$

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

${\displaystyle \int \tanh ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\tanh ^{n-1}ax+\int \tanh ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}$

${\displaystyle \int \coth x\,dx=\ln |\sinh x|+C,{\text{ for }}x\neq 0}$ ${\displaystyle \int \coth ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\coth ^{n-1}ax+\int \coth ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}$

${\displaystyle \int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C}$ ${\displaystyle \int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C,{\text{ for }}x\neq 0}$

Integral melibatkan fungsi hiperbolik sinus dan kosinus

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

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

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

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

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

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

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

Integral melibatkan fungsi hiperbolik dan trigonometri

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

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

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

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