Daftar integral dari fungsi hiperbolik: Perbedaan antara revisi

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

 

 

<math>\int\sinh ax\,dx = \frac{1}{a}\cosh ax+C\,</math>

 

: juga: <math>\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{)}\,</math>

 

<math>\int\frac{dx}{\sinh ax} = \frac{1}{a} \ln\left|\tanh\frac{ax}{2}\right|+C\,</math>

 

: <math>\int\frac{dx}{\sinh ax} = \frac{1}{2a} \ln\left|\frac{\cosh ax - 1}{\cosh ax + 1}\right|+C\,</math>

 

<math>\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{)}\,</math>

<math>\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{)}\,</math>

 

 

<math>\int\cosh ax\,dx = \frac{1}{a}\sinh ax+C\,</math>

 

: juga: <math>\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{)}\,</math>

 

<math>\int\frac{dx}{\cosh ax} = \frac{2}{a} \arctan e^{ax}+C\,</math>

 

: juga: <math>\int\frac{dx}{\cosh ax} = \frac{1}{a} \arctan (\sinh ax)+C\,</math>

 

<math>\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{)}\,</math>

== Integral lain-lain ==

 

 

<math>\int \tanh x \, dx = \ln \cosh x + C</math>

<math>\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{)}\,</math>

 

<math>\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{)}\,</math>

 

<math>\int \operatorname{sech}\,x \, dx = \arctan\,(\sinh x) + C</math>

 

 

<math>\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{)}\,</math>

: <math>\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{)}\,</math>

 

 

<math>\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\,</math>

{{Daftar integral}}

 

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