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Baris 12: |
Baris 12: |
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: 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> |
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: 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> |
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<math>\int\frac{dx}{\sinh ax} = \frac{1}{a} \ln\left|\tanh\frac{ax}{2}\right|+C\,</math> |
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<math>\int\frac{dx}{\sinh ax} = \frac{1}{a} \ln\left|\tanh\frac{ax}{2}\right|+C\,</math> |
Baris 21: |
Baris 20: |
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: <math>\int\frac{dx}{\sinh ax} = \frac{1}{2a} \ln\left|\frac{\cosh ax - 1}{\cosh ax + 1}\right|+C\,</math> |
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: <math>\int\frac{dx}{\sinh ax} = \frac{1}{2a} \ln\left|\frac{\cosh ax - 1}{\cosh ax + 1}\right|+C\,</math> |
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<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> |
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<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> |
Baris 38: |
Baris 36: |
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: 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> |
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: 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> |
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<math>\int\frac{dx}{\cosh ax} = \frac{2}{a} \arctan e^{ax}+C\,</math> |
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<math>\int\frac{dx}{\cosh ax} = \frac{2}{a} \arctan e^{ax}+C\,</math> |
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: juga: <math>\int\frac{dx}{\cosh ax} = \frac{1}{a} \arctan (\sinh ax)+C\,</math> |
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: juga: <math>\int\frac{dx}{\cosh ax} = \frac{1}{a} \arctan (\sinh ax)+C\,</math> |
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<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> |
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<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> |
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.
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