双曲函数积分表

以下是部份双曲函数的积分表(书写时省略了不定积分结果中都含有的任意常数Cn) ∫ sinh ⁡ c x d x = 1 c cosh ⁡ c x {\displaystyle \int \sinh cx\,dx={\frac {1}{c}}\cosh cx} ∫ cosh ⁡ c x d x = 1 c sinh ⁡ c x {\displaystyle \int \cosh cx\,dx={\frac {1}{c}}\sinh cx} ∫ sinh 2 ⁡ c x d x = 1 4 c sinh ⁡ 2 c x − x 2 {\displaystyle \int \sinh ^{2}cx\,dx={\frac {1}{4c}}\sinh 2cx-{\frac {x}{2}}} ∫ cosh 2 ⁡ c x d x = 1 4 c sinh ⁡ 2 c x + x 2 {\displaystyle \int \cosh ^{2}cx\,dx={\frac {1}{4c}}\sinh 2cx+{\frac {x}{2}}} ∫ sinh n ⁡ c x d x = 1 c n sinh n − 1 ⁡ c x cosh ⁡ c x − n − 1 n ∫ sinh n − 2 ⁡ c x d x (for  n > 0 ) {\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{(for }}n>0{\mbox{)}}} ∫ sinh n ⁡ c x d x = 1 c ( n + 1 ) sinh n + 1 ⁡ c x cosh ⁡ c x − n + 2 n + 1 ∫ sinh n + 2 ⁡ c x d x (for  n < 0 ,  n ≠ − 1 ) {\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{(for }}n<0{\mbox{, }}n\neq -1{\mbox{)}}} ∫ cosh n ⁡ c x d x = 1 c n sinh ⁡ c x cosh n − 1 ⁡ c x + n − 1 n ∫ cosh n − 2 ⁡ c x d x (for  n > 0 ) {\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{(for }}n>0{\mbox{)}}} ∫ cosh n ⁡ c x d x = − 1 c ( n + 1 ) sinh ⁡ c x cosh n + 1 ⁡ c x − n + 2 n + 1 ∫ cosh n + 2 ⁡ c x d x (for  n < 0 ,  n ≠ − 1 ) {\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{(for }}n<0{\mbox{, }}n\neq -1{\mbox{)}}} ∫ d x sinh ⁡ c x = 1 c ln ⁡ | tanh ⁡ c x 2 | {\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|\tanh {\frac {cx}{2}}\right|} ∫ d x sinh ⁡ c x = 1 c ln ⁡ | cosh ⁡ c x − 1 sinh ⁡ c x | {\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\sinh cx}}\right|} ∫ d x sinh ⁡ c x = 1 c ln ⁡ | sinh ⁡ c x cosh ⁡ c x + 1 | {\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\sinh cx}{\cosh cx+1}}\right|} ∫ d x sinh ⁡ c x = 1 2 c ln ⁡ | cosh ⁡ c x − 1 cosh ⁡ c x + 1 | {\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{2c}}\ln \left|{\frac {\cosh cx-1}{\cosh cx+1}}\right|} ∫ d x cosh ⁡ c x = 2 c arctan ⁡ e c x {\displaystyle \int {\frac {dx}{\cosh cx}}={\frac {2}{c}}\arctan e^{cx}} ∫ d x sinh n ⁡ c x = cosh ⁡ c x c ( n − 1 ) sinh n − 1 ⁡ c x − n − 2 n − 1 ∫ d x sinh n − 2 ⁡ c x (for  n ≠ 1 ) {\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{(for }}n\neq 1{\mbox{)}}} ∫ d x cosh n ⁡ c x = sinh ⁡ c x c ( n − 1 ) cosh n − 1 ⁡ c x + n − 2 n − 1 ∫ d x cosh n − 2 ⁡ c x (for  n ≠ 1 ) {\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{(for }}n\neq 1{\mbox{)}}} ∫ cosh n ⁡ c x sinh m ⁡ c x d x = cosh n − 1 ⁡ c x c ( n − m ) sinh m − 1 ⁡ c x + n − 1 n − m ∫ cosh n − 2 ⁡ c x sinh m ⁡ c x d x (for  m ≠ n ) {\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{(for }}m\neq n{\mbox{)}}} ∫ cosh n ⁡ c x sinh m ⁡ c x d x = − cosh n + 1 ⁡ c x c ( m − 1 ) sinh m − 1 ⁡ c x + n − m + 2 m − 1 ∫ cosh n ⁡ c x sinh m − 2 ⁡ c x d x (for  m ≠ 1 ) {\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{(for }}m\neq 1{\mbox{)}}} ∫ cosh n ⁡ c x sinh m ⁡ c x d x = − cosh n − 1 ⁡ c x c ( m − 1 ) sinh m − 1 ⁡ c x + n − 1 m − 1 ∫ cosh n − 2 ⁡ c x sinh m − 2 ⁡ c x d x (for  m ≠ 1 ) {\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{(for }}m\neq 1{\mbox{)}}} ∫ sinh m ⁡ c x cosh n ⁡ c x d x = sinh m − 1 ⁡ c x c ( m − n ) cosh n − 1 ⁡ c x + m − 1 m − n ∫ sinh m − 2 ⁡ c x cosh n ⁡ c x d x (for  m ≠ n ) {\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{(for }}m\neq n{\mbox{)}}} ∫ sinh m ⁡ c x cosh n ⁡ c x d x = sinh m + 1 ⁡ c x c ( n − 1 ) cosh n − 1 ⁡ c x + m − n + 2 n − 1 ∫ sinh m ⁡ c x cosh n − 2 ⁡ c x d x (for  n ≠ 1 ) {\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{(for }}n\neq 1{\mbox{)}}} ∫ sinh m ⁡ c x cosh n ⁡ c x d x = − sinh m − 1 ⁡ c x c ( n − 1 ) cosh n − 1 ⁡ c x + m − 1 n − 1 ∫ sinh m − 2 ⁡ c x cosh n − 2 ⁡ c x d x (for  n ≠ 1 ) {\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{(for }}n\neq 1{\mbox{)}}} ∫ x sinh ⁡ c x d x = 1 c x cosh ⁡ c x − 1 c 2 sinh ⁡ c x {\displaystyle \int x\sinh cx\,dx={\frac {1}{c}}x\cosh cx-{\frac {1}{c^{2}}}\sinh cx} ∫ x cosh ⁡ c x d x = 1 c x sinh ⁡ c x − 1 c 2 cosh ⁡ c x {\displaystyle \int x\cosh cx\,dx={\frac {1}{c}}x\sinh cx-{\frac {1}{c^{2}}}\cosh cx} ∫ tanh ⁡ c x d x = 1 c ln ⁡ | cosh ⁡ c x | {\displaystyle \int \tanh cx\,dx={\frac {1}{c}}\ln |\cosh cx|} ∫ coth ⁡ c x d x = 1 c ln ⁡ | sinh ⁡ c x | {\displaystyle \int \coth cx\,dx={\frac {1}{c}}\ln |\sinh cx|} ∫ tanh n ⁡ c x d x = − 1 c ( n − 1 ) tanh n − 1 ⁡ c x + ∫ tanh n − 2 ⁡ c x d x (for  n ≠ 1 ) {\displaystyle \int \tanh ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\tanh ^{n-1}cx+\int \tanh ^{n-2}cx\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} ∫ coth n ⁡ c x d x = − 1 c ( n − 1 ) coth n − 1 ⁡ c x + ∫ coth n − 2 ⁡ c x d x (for  n ≠ 1 ) {\displaystyle \int \coth ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\coth ^{n-1}cx+\int \coth ^{n-2}cx\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} ∫ sinh ⁡ b x sinh ⁡ c x d x = 1 b 2 − c 2 ( b sinh ⁡ c x cosh ⁡ b x − c cosh ⁡ c x sinh ⁡ b x ) (for  b 2 ≠ c 2 ) {\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{(for }}b^{2}\neq c^{2}{\mbox{)}}} ∫ cosh ⁡ b x cosh ⁡ c x d x = 1 b 2 − c 2 ( b sinh ⁡ b x cosh ⁡ c x − c sinh ⁡ c x cosh ⁡ b x ) (for  b 2 ≠ c 2 ) {\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{(for }}b^{2}\neq c^{2}{\mbox{)}}} ∫ cosh ⁡ b x sinh ⁡ c x d x = 1 b 2 − c 2 ( b sinh ⁡ b x sinh ⁡ c x − c cosh ⁡ b x cosh ⁡ c x ) (for  b 2 ≠ c 2 ) {\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{(for }}b^{2}\neq c^{2}{\mbox{)}}} ∫ sinh ⁡ ( a x + b ) sin ⁡ ( c x + d ) d x = a a 2 + c 2 cosh ⁡ ( a x + b ) sin ⁡ ( c x + d ) − c a 2 + c 2 sinh ⁡ ( a x + b ) cos ⁡ ( c x + d ) {\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)} ∫ sinh ⁡ ( a x + b ) cos ⁡ ( c x + d ) d x = a a 2 + c 2 cosh ⁡ ( a x + b ) cos ⁡ ( c x + d ) + c a 2 + c 2 sinh ⁡ ( a x + b ) sin ⁡ ( c x + 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)} ∫ cosh ⁡ ( a x + b ) sin ⁡ ( c x + d ) d x = a a 2 + c 2 sinh ⁡ ( a x + b ) sin ⁡ ( c x + d ) − c a 2 + c 2 cosh ⁡ ( a x + b ) cos ⁡ ( c x + 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)} ∫ cosh ⁡ ( a x + b ) cos ⁡ ( c x + d ) d x = a a 2 + c 2 sinh ⁡ ( a x + b ) cos ⁡ ( c x + d ) + c a 2 + c 2 cosh ⁡ ( a x + b ) sin ⁡ ( c x + 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)}