Daftar integral (antiderivatif) dari fungsi eksponensial. Untuk daftar lengkap fungsi integral, lihat Tabel integral.
Dalam semua rumus, konstanta a diasumsikan bukan nol.
Integral tak tentu
Integral tak tentu adalah fungsi-fungsi antiderivatif. Sebuah konstanta (yaitu konstanta integrasi) dapat ditambahkan pada sisi kanan dari rumus ini, tetapi tidak dituliskan di sini demi kesederhanaan.
Integral melibatkan hanya fungsi eksponensial
![{\displaystyle \int \mathrm {f} '(x)e^{f(x)}\;\mathrm {d} x=e^{f(x)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9830384c3f0e241353850a4cbbe68682da4e5eb)
![{\displaystyle \int e^{cx}\;\mathrm {d} x={\frac {1}{c}}e^{cx}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8a2d8d0ed5abc7ccb0bd31adf4274be7dd54ea)
for ![{\displaystyle a>0,\ a\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c27d65d4a22ff9cb56ae8562f0ba1ee2b4a7f120)
Integral melibatkan fungsi eksponensial dan pangkat
![{\displaystyle \int xe^{cx}\;\mathrm {d} x={\frac {e^{cx}}{c^{2}}}(cx-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/500198d16fb049f067c35b7e7cb88d132fcb7783)
- \int xe^{-cx}\; \mathrm{d}x =x \frac{1}{-c}e^{-cx}
![{\displaystyle \int x^{2}e^{cx}\;\mathrm {d} x=e^{cx}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e267454873adf7adc1827062f44cad4d17488c2)
![{\displaystyle \int x^{n}e^{cx}\;\mathrm {d} x={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}\mathrm {d} x=\left({\frac {\partial }{\partial c}}\right)^{n}{\frac {e^{cx}}{c}}=e^{cx}\sum _{i=0}^{n}(-1)^{i}\,{\frac {n!}{(n-i)!\,c^{i+1}}}\,x^{n-i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d21115350c6677018372dd7d2c39ea43b3f8b25)
![{\displaystyle \int {\frac {e^{cx}}{x}}\;\mathrm {d} x=\ln |x|+\sum _{n=1}^{\infty }{\frac {(cx)^{n}}{n\cdot n!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c73c96bbd063b40cd6bb1ced9c96f8d98a5de8f)
![{\displaystyle \int {\frac {e^{cx}}{x^{n}}}\;\mathrm {d} x={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}}{x^{n-1}}}\,\mathrm {d} x\right)\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3edd292ca8743cc942cd94d98458d3b720a085ce)
Integral melibatkan fungsi eksponensial dan trigonometri
![{\displaystyle \int e^{cx}\sin bx\;\mathrm {d} x={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)={\frac {e^{cx}}{\sqrt {c^{2}+b^{2}}}}\sin(bx-\phi )\qquad \cos(\phi )={\frac {c}{\sqrt {c^{2}+b^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9d43ee9c7e281b81f7443f281a84827783b4cb)
![{\displaystyle \int e^{cx}\cos bx\;\mathrm {d} x={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)={\frac {e^{cx}}{\sqrt {c^{2}+b^{2}}}}\cos(bx-\phi )\qquad \cos(\phi )={\frac {c}{\sqrt {c^{2}+b^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80297085c967d1821d9363ac9811f79cdc099bf6)
![{\displaystyle \int e^{cx}\sin ^{n}x\;\mathrm {d} x={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\sin ^{n-2}x\;\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f62ae0a81bf35f4e80680484429d46032ac888)
![{\displaystyle \int e^{cx}\cos ^{n}x\;\mathrm {d} x={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\cos ^{n-2}x\;\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a91207df8692dd303615aee89b17de3c86b72079)
Integral melibatkan fungsi kesalahan
![{\displaystyle \int e^{cx}\ln x\;\mathrm {d} x={\frac {1}{c}}\left(e^{cx}\ln |x|-\operatorname {Ei} \,(cx)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09efcfeec0db1b4feed49557386f8b622d7a0c76)
![{\displaystyle \int xe^{cx^{2}}\;\mathrm {d} x={\frac {1}{2c}}\;e^{cx^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eaad8f6cee041e8e3ce3a611f8ebaf9df994ce1b)
(
adalah suatu fungsi error)
![{\displaystyle \int xe^{-cx^{2}}\;\mathrm {d} x=-{\frac {1}{2c}}e^{-cx^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30467d4ac29c992b8fd62db75447783e08e374df)
![{\displaystyle \int {\frac {e^{-x^{2}}}{x^{2}}}\;\mathrm {d} x=-{\frac {e^{-x^{2}}}{x}}-{\sqrt {\pi }}\mathrm {erf} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/111d80f6c931225a9e5b339dfb012fba6c24fca5)
![{\displaystyle \int {{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}\;\mathrm {d} x={\frac {1}{2}}\left(\operatorname {erf} \,{\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d617e3f0726b343188ff777820169425d7c0e20)
Integral lain-lain
- di mana
![{\displaystyle c_{2j}={\frac {1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}}}={\frac {(2j)\,!}{j!\,2^{2j+1}}}\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90dd5e5695126e16ae330b936f2ae1abc90fcf3a)
- (Perhatikan bahwa nilai ekspresi ini independen atau tidak tergantung dari nilai
, karena itu tidak muncul dalam integral.)
- di mana
![{\displaystyle a_{mn}={\begin{cases}1&{\text{jika }}n=0,\\{\frac {1}{n!}}&{\text{jika }}m=1,\\{\frac {1}{n}}\sum _{j=1}^{n}ja_{m,n-j}a_{m-1,j-1}&{\text{selainnya}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca007b6e4e314cd0a98c234e79fbb0d316ee2f00)
- dan
adalah fungsi gamma
ketika
,
, dan ![{\displaystyle ae^{\lambda x}+b>0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3728b822a9eafe02b95482cf00c7182f01eb43)
ketika
,
, dan ![{\displaystyle ae^{\lambda x}+b>0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3728b822a9eafe02b95482cf00c7182f01eb43)
Integral tertentu
untuk
, yang merupakan rata-rata logaritme
![{\displaystyle \int _{0}^{\infty }e^{ax}\,\mathrm {d} x={\frac {1}{-a}}\quad (\operatorname {Re} (a)<0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5aad5beee968042e51edb9eb2715ee08c9c0939)
(Integral Gaussian)
![{\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}\,\mathrm {d} x={\sqrt {\pi \over a}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/726b7008c42f0da6d3d9b4b3df8bcb2aac6b4ac6)
(lihat Integral suatu fungsi Gaussian)
![{\displaystyle \int _{-\infty }^{\infty }xe^{-a(x-b)^{2}}\,\mathrm {d} x=b{\sqrt {\frac {\pi }{a}}}\quad (\operatorname {Re} (a)>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72d0bfdbd7c4375e0b446eff356b0021dafafec8)
![{\displaystyle \int _{-\infty }^{\infty }xe^{-ax^{2}+bx}\,\mathrm {d} x={\frac {{\sqrt {\pi }}b}{2a^{3/2}}}e^{\frac {b^{2}}{4a}}\quad (\operatorname {Re} (a)>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b6588fd8b110ae016ffaf44b6a3966524cf9bd)
![{\displaystyle \int _{-\infty }^{\infty }x^{2}e^{-ax^{2}}\,\mathrm {d} x={\frac {1}{2}}{\sqrt {\pi \over a^{3}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/034c82717067aa92e4672ae9ca69d5e317e12f22)
![{\displaystyle \int _{-\infty }^{\infty }x^{2}e^{-ax^{2}-bx}\,\mathrm {d} x={\frac {{\sqrt {\pi }}(2a+b^{2})}{4a^{5/2}}}e^{\frac {b^{2}}{4a}}\quad (\operatorname {Re} (a)>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f598403b83aa19ad94d4ebf0e3f72c8db2331b4e)
![{\displaystyle \int _{-\infty }^{\infty }x^{3}e^{-ax^{2}+bx}\,\mathrm {d} x={\frac {{\sqrt {\pi }}(6a+b^{2})b}{8a^{7/2}}}e^{\frac {b^{2}}{4a}}\quad (\operatorname {Re} (a)>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb6f9ebccf968f9a78bcc676067c3148a11824d)
(!! merupakan faktorial ganda)
![{\displaystyle \int _{0}^{\infty }x^{n}e^{-ax}\,\mathrm {d} x={\begin{cases}{\frac {\Gamma (n+1)}{a^{n+1}}}&(n>-1,a>0)\\{\frac {n!}{a^{n+1}}}&(n=0,1,2,\ldots ,a>0)\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e56ff6ccc4fcb5b354ad33de0cfda66500454c6)
![{\displaystyle \int _{0}^{1}x^{n}e^{-ax}\,\mathrm {d} x={\frac {n!}{a^{n+1}}}\left[1-e^{-a}\sum _{i=0}^{n}{\frac {a^{i}}{i!}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8646de30e21caf89876f564001c275877a219bf7)
![{\displaystyle \int _{0}^{\infty }e^{-ax^{b}}dx={\frac {1}{b}}\ a^{-{\frac {1}{b}}}\,\Gamma \left({\frac {1}{b}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ec6200b217ea3c5dcab17e1f1c10d81e06153a)
![{\displaystyle \int _{0}^{\infty }x^{n}e^{-ax^{b}}dx={\frac {1}{b}}\ a^{-{\frac {n+1}{b}}}\,\Gamma \left({\frac {n+1}{b}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecca8baa65309a7a6396807e9d842a43c8d1c96e)
![{\displaystyle \int _{0}^{\infty }e^{-ax}\sin bx\,\mathrm {d} x={\frac {b}{a^{2}+b^{2}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/116b9485eb78a3df65cac57937643c8091ddfb7e)
![{\displaystyle \int _{0}^{\infty }e^{-ax}\cos bx\,\mathrm {d} x={\frac {a}{a^{2}+b^{2}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3823f932a06e10b125bfe342f2baa1023cdaa350)
![{\displaystyle \int _{0}^{\infty }xe^{-ax}\sin bx\,\mathrm {d} x={\frac {2ab}{(a^{2}+b^{2})^{2}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e3b468a59eb2d4b6daa62060b7e3f52ad3da2f)
![{\displaystyle \int _{0}^{\infty }xe^{-ax}\cos bx\,\mathrm {d} x={\frac {a^{2}-b^{2}}{(a^{2}+b^{2})^{2}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee55434eae15e366659f77b386ed7056af99478a)
(
adalah modifikasi fungsi Bessel dari jenis pertama)
![{\displaystyle \int _{0}^{2\pi }e^{x\cos \theta +y\sin \theta }d\theta =2\pi I_{0}\left({\sqrt {x^{2}+y^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc7da0077239149468cbcc5eb3576109c8d0d4d)
Pranala luar
- Wolfram Mathematica Online Integrator
- V. H. Moll, The Integrals in Gradshteyn and Ryzhik