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List of integrals of hyperbolic functions
From Wikipedia, the free encyclopedia
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals.
In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.
Integrals involving only hyperbolic sine functions[edit]
- ∫ sinh a x d x = 1 a cosh a x + C {\displaystyle \int \sinh ax\,dx={\frac {1}{a}}\cosh ax+C}
- ∫ sinh 2 a x d x = 1 4 a sinh 2 a x − x 2 + C {\displaystyle \int \sinh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax-{\frac {x}{2}}+C}
- ∫ sinh n a x d x = { 1 a n ( sinh n − 1 a x ) ( cosh a x ) − n − 1 n ∫ sinh n − 2 a x d x , n > 0 1 a ( n + 1 ) ( sinh n + 1 a x ) ( cosh a x ) − n + 2 n + 1 ∫ sinh n + 2 a x d x , n < 0 , n ≠ − 1 {\displaystyle \int \sinh ^{n}ax\,dx={\begin{cases}{\frac {1}{an}}(\sinh ^{n-1}ax)(\cosh ax)-{\frac {n-1}{n}}\displaystyle \int \sinh ^{n-2}ax\,dx,&n>0\\{\frac {1}{a(n+1)}}(\sinh ^{n+1}ax)(\cosh ax)-{\frac {n+2}{n+1}}\displaystyle \int \sinh ^{n+2}ax\,dx,&n<0,n\neq -1\end{cases}}}
- ∫ d x sinh a x = 1 a ln | tanh a x 2 | + C = 1 a ln | cosh a x − 1 sinh a x | + C = 1 a ln | sinh a x cosh a x + 1 | + C = 1 2 a ln | cosh a x − 1 cosh a x + 1 | + C {\displaystyle {\begin{aligned}\int {\frac {dx}{\sinh ax}}&={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2}}\right|+C\\&={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\sinh ax}}\right|+C\\&={\frac {1}{a}}\ln \left|{\frac {\sinh ax}{\cosh ax+1}}\right|+C\\&={\frac {1}{2a}}\ln \left|{\frac {\cosh ax-1}{\cosh ax+1}}\right|+C\end{aligned}}}
- ∫ d x sinh n a x = − cosh a x a ( n − 1 ) sinh n − 1 a x − n − 2 n − 1 ∫ d x sinh n − 2 a x (for n ≠ 1 ) {\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{)}}}
- ∫ x sinh a x d x = 1 a x cosh a x − 1 a 2 sinh a x + C {\displaystyle \int x\sinh ax\,dx={\frac {1}{a}}x\cosh ax-{\frac {1}{a^{2}}}\sinh ax+C}
- ∫ ( sinh a x ) ( sinh b x ) d x = 1 a 2 − b 2 ( a ( sinh b x ) ( cosh a x ) − b ( cosh b x ) ( sinh a x ) ) + C (for a 2 ≠ b 2 ) {\displaystyle \int (\sinh ax)(\sinh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh bx)(\cosh ax)-b(\cosh bx)(\sinh ax){\big )}+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}}
Integrals involving only hyperbolic cosine functions[edit]
- ∫ cosh a x d x = 1 a sinh a x + C {\displaystyle \int \cosh ax\,dx={\frac {1}{a}}\sinh ax+C}
- ∫ cosh 2 a x d x = 1 4 a sinh 2 a x + x 2 + C {\displaystyle \int \cosh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax+{\frac {x}{2}}+C}
- ∫ cosh n a x d x = { 1 a n ( sinh a x ) ( cosh n − 1 a x ) + n − 1 n ∫ cosh n − 2 a x d x , n > 0 − 1 a ( n + 1 ) ( sinh a x ) ( cosh n + 1 a x ) + n + 2 n + 1 ∫ cosh n + 2 a x d x , n < 0 , n ≠ − 1 {\displaystyle \int \cosh ^{n}ax\,dx={\begin{cases}{\frac {1}{an}}(\sinh ax)(\cosh ^{n-1}ax)+{\frac {n-1}{n}}\displaystyle \int \cosh ^{n-2}ax\,dx,&n>0\\-{\frac {1}{a(n+1)}}(\sinh ax)(\cosh ^{n+1}ax)+{\frac {n+2}{n+1}}\displaystyle \int \cosh ^{n+2}ax\,dx,&n<0,n\neq -1\end{cases}}}
- ∫ d x cosh a x = 2 a arctan e a x + C = 1 a arctan ( sinh a x ) + C {\displaystyle {\begin{aligned}\int {\frac {dx}{\cosh ax}}&={\frac {2}{a}}\arctan e^{ax}+C\\&={\frac {1}{a}}\arctan(\sinh ax)+C\end{aligned}}}
- ∫ d x cosh n a x = sinh a x a ( n − 1 ) cosh n − 1 a x + n − 2 n − 1 ∫ d x cosh n − 2 a x (for n ≠ 1 ) {\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{)}}}
- ∫ x cosh a x d x = 1 a x sinh a x − 1 a 2 cosh a x + C {\displaystyle \int x\cosh ax\,dx={\frac {1}{a}}x\sinh ax-{\frac {1}{a^{2}}}\cosh ax+C}
- ∫ x 2 cosh a x d x = − 2 x cosh a x a 2 + ( x 2 a + 2 a 3 ) sinh a x + 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}
- ∫ ( cosh a x ) ( cosh b x ) d x = 1 a 2 − b 2 ( a ( sinh a x ) ( cosh b x ) − b ( sinh b x ) ( cosh a x ) ) + C (for a 2 ≠ b 2 ) {\displaystyle \int (\cosh ax)(\cosh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh ax)(\cosh bx)-b(\sinh bx)(\cosh ax){\big )}+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}}
- ∫ d x 1 + cosh ( a x ) = 2 a 1 1 + e − a x + C {\displaystyle \int {\frac {dx}{1+\cosh(ax)}}={\frac {2}{a}}{\frac {1}{1+e^{-ax}}}+C\quad } or 2 a {\displaystyle {\frac {2}{a}}} times The Logistic Function
Integrals of hyperbolic tangent, cotangent, secant, cosecant functions[edit]
- ∫ tanh x d x = ln cosh x + C {\displaystyle \int \tanh x\,dx=\ln \cosh x+C}
- ∫ tanh 2 a x d x = x − tanh a x a + C {\displaystyle \int \tanh ^{2}ax\,dx=x-{\frac {\tanh ax}{a}}+C}
- ∫ tanh n a x d x = − 1 a ( n − 1 ) tanh n − 1 a x + ∫ tanh n − 2 a x d x (for n ≠ 1 ) {\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{)}}}
- ∫ coth x d x = ln | sinh x | + C , for x ≠ 0 {\displaystyle \int \coth x\,dx=\ln |\sinh x|+C,{\text{ for }}x\neq 0}
- ∫ coth n a x d x = − 1 a ( n − 1 ) coth n − 1 a x + ∫ coth n − 2 a x d x (for n ≠ 1 ) {\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{)}}}
- ∫ sech x d x = arctan ( sinh x ) + C {\displaystyle \int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C}
- ∫ csch x d x = ln | tanh x 2 | + C = ln | coth x − csch x | + C , for x ≠ 0 {\displaystyle \int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C=\ln \left|\coth {x}-\operatorname {csch} {x}\right|+C,{\text{ for }}x\neq 0}
Integrals involving hyperbolic sine and cosine functions[edit] Integrals involving hyperbolic and trigonometric functions[edit]
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