Danh sách tích phân với hàm hypebolic

Dưới đây là danh sách tích phân với hàm hypebolic.

${\displaystyle \int \sinh cx\,dx={\frac {1}{c}\cosh cx}$

${\displaystyle \int \cosh cx\,dx={\frac {1}{c}\sinh cx}$

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

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

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

hay: ${\displaystyle \int \sinh ^{n}cx\,dx={\frac {1}{c(n+1)}\sinh ^{n+1}cx\cosh cx-{\frac {n+2}{n+1}\int \sinh ^{n+2}cx\,dx\qquad {\mbox{(}n<0{\mbox{, }n\neq -1{\mbox{)}$

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

hay: ${\displaystyle \int \cosh ^{n}cx\,dx=-{\frac {1}{c(n+1)}\sinh cx\cosh ^{n+1}cx-{\frac {n+2}{n+1}\int \cosh ^{n+2}cx\,dx\qquad {\mbox{(}n<0{\mbox{, }n\neq -1{\mbox{)}$

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

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

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

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

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

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

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

${\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}dx={\frac {\cosh ^{n-1}cx}{c(n-m)\sinh ^{m-1}cx}+{\frac {n-1}{n-m}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m}cx}dx\qquad {\mbox{(}m\neq n{\mbox{)}$

hay: ${\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}dx=-{\frac {\cosh ^{n+1}cx}{c(m-1)\sinh ^{m-1}cx}+{\frac {n-m+2}{m-1}\int {\frac {\cosh ^{n}cx}{\sinh ^{m-2}cx}dx\qquad {\mbox{(}m\neq 1{\mbox{)}$

hay: ${\displaystyle \int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}dx=-{\frac {\cosh ^{n-1}cx}{c(m-1)\sinh ^{m-1}cx}+{\frac {n-1}{m-1}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m-2}cx}dx\qquad {\mbox{(}m\neq 1{\mbox{)}$

${\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}dx={\frac {\sinh ^{m-1}cx}{c(m-n)\cosh ^{n-1}cx}+{\frac {m-1}{m-n}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n}cx}dx\qquad {\mbox{(}m\neq n{\mbox{)}$

hay: ${\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}dx={\frac {\sinh ^{m+1}cx}{c(n-1)\cosh ^{n-1}cx}+{\frac {m-n+2}{n-1}\int {\frac {\sinh ^{m}cx}{\cosh ^{n-2}cx}dx\qquad {\mbox{(}n\neq 1{\mbox{)}$

hay: ${\displaystyle \int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}dx=-{\frac {\sinh ^{m-1}cx}{c(n-1)\cosh ^{n-1}cx}+{\frac {m-1}{n-1}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n-2}cx}dx\qquad {\mbox{(}n\neq 1{\mbox{)}$

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

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

${\displaystyle \int \tanh cx\,dx={\frac {1}{c}\ln |\cosh cx|}$

${\displaystyle \int \coth cx\,dx={\frac {1}{c}\ln |\sinh cx|}$

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

${\displaystyle \int \coth ^{n}cx\,dx=-{\frac {1}{c(n-1)}\coth ^{n-1}cx+\int \coth ^{n-2}cx\,dx\qquad {\mbox{(}n\neq 1{\mbox{)}$

${\displaystyle \int \sinh bx\sinh cx\,dx={\frac {1}{b^{2}-c^{2}(b\sinh cx\cosh bx-c\cosh cx\sinh bx)\qquad {\mbox{(}b^{2}\neq c^{2}{\mbox{)}$

${\displaystyle \int \cosh bx\cosh cx\,dx={\frac {1}{b^{2}-c^{2}(b\sinh bx\cosh cx-c\sinh cx\cosh bx)\qquad {\mbox{(}b^{2}\neq c^{2}{\mbox{)}$

${\displaystyle \int \cosh bx\sinh cx\,dx={\frac {1}{b^{2}-c^{2}(b\sinh bx\sinh cx-c\cosh bx\cosh cx)\qquad {\mbox{(}b^{2}\neq c^{2}{\mbox{)}$

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

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

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

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

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