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Dalam semua rumus, konstanta ''a'' diasumsikan bukan nol, dan ''C'' melambangkan [[konstanta integrasi]]. |
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Dalam semua rumus, konstanta ''a'' diasumsikan bukan nol, dan ''C'' melambangkan [[konstanta integrasi]]. |
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==Integral melibatkan hanya fungsi hiperbolik sinus== |
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== Integral melibatkan hanya fungsi hiperbolik sinus == |
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<math>\int\sinh ax\,dx = \frac{1}{a}\cosh ax+C\,</math> |
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<math>\int\sinh ax\,dx = \frac{1}{a}\cosh ax+C\,</math> |
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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> |
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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> |
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<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> |
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<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> |
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==Integral melibatkan hanya fungsi hiperbolik kosinus== |
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== Integral melibatkan hanya fungsi hiperbolik kosinus == |
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<math>\int\cosh ax\,dx = \frac{1}{a}\sinh ax+C\,</math> |
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<math>\int\cosh ax\,dx = \frac{1}{a}\sinh ax+C\,</math> |
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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> |
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== Integral lain-lain == |
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== Integral lain-lain == |
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===Integral fungsi hiperbolik tangen, kotangen, sekan, kosekan === |
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=== Integral fungsi hiperbolik tangen, kotangen, sekan, kosekan === |
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<math>\int \tanh x \, dx = \ln \cosh x + C</math> |
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<math>\int \tanh x \, dx = \ln \cosh x + C</math> |
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<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> |
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<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> |
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<math>\int \coth x \, dx = \ln| \sinh x | + C , \text{ for } x \neq 0 </math> |
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<math>\int \coth x \, dx = \ln| \sinh x | + C, \text{ for } x \neq 0 </math> |
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<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> |
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<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> |
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<math>\int \operatorname{sech}\,x \, dx = \arctan\,(\sinh x) + C</math> |
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<math>\int \operatorname{sech}\,x \, dx = \arctan\,(\sinh x) + C</math> |
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<math>\int \operatorname{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C , \text{ for } x \neq 0 </math> |
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<math>\int \operatorname{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C, \text{ for } x \neq 0 </math> |
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===Integral melibatkan fungsi hiperbolik sinus dan kosinus === |
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=== Integral melibatkan fungsi hiperbolik sinus dan kosinus === |
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<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> |
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<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> |
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: <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> |
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: <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> |
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===Integral melibatkan fungsi hiperbolic dan trigonometri === |
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=== Integral melibatkan fungsi hiperbolik dan trigonometri === |
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<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> |
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<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> |
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{{Daftar integral}} |
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{{Daftar integral}} |
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[[Category:Eksponensial]] |
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[[Kategori:Eksponensial]] |
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[[Category:Integral]] |
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[[Kategori:Integral]] |
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[[Kategori:Kalkulus]] |
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[[Kategori:Kalkulus]] |
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.
Integral melibatkan hanya fungsi hiperbolik sinus[sunting | sunting sumber]
- juga:
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8cd20cfe958b17f3c2e1ff5ee54d5608ddc9f2)
- juga:
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\sinh ax}}\right|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8d31337ed2598b6933cb5ea53b9a77c44e139c)
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\sinh ax}{\cosh ax+1}}\right|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9abc694d34204ecd7ccd4af1576834fe16bf8387)
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{2a}}\ln \left|{\frac {\cosh ax-1}{\cosh ax+1}}\right|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed2c5e9f0572476dce14c6272a758887d5a5bc7a)
Integral melibatkan hanya fungsi hiperbolik kosinus[sunting | sunting sumber]
- juga:
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b00d7b9291030d9ecbdb65ef3180a667a686a86c)
- juga:
![{\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {1}{a}}\arctan(\sinh ax)+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50ddb5b6da0e0038f4fa6249924121949b4387d3)
Integral fungsi hiperbolik tangen, kotangen, sekan, kosekan[sunting | sunting sumber]
Integral melibatkan fungsi hiperbolik sinus dan kosinus[sunting | sunting sumber]
- juga:
![{\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx=-{\frac {\cosh ^{n+1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}ax}{\sinh ^{m-2}ax}}dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7572b1393192e541db41e1955c55e899b2398e0)
![{\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx=-{\frac {\cosh ^{n-1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m-2}ax}}dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a23be9bfbc718da7eb953938ab893daa8606cd5)
![{\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx={\frac {\sinh ^{m-1}ax}{a(m-n)\cosh ^{n-1}ax}}+{\frac {m-1}{n-m}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n}ax}}dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac64edb7da52a49884dd2c9521709389acd8b92)
![{\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx={\frac {\sinh ^{m+1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}ax}{\cosh ^{n-2}ax}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46658b84dd546f7e559b1da9860b81095aa2a261)
![{\displaystyle \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{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2bc949c723b76c32dfaafb96807c3a41dd75cf)
Integral melibatkan fungsi hiperbolik dan trigonometri[sunting | sunting sumber]