De la Wikipedia, enciclopedia liberă
Următorul articol este o listă de integrale (primitive) de funcții exponențiale. Pentru o listă cu mai multe integrale, vezi tabel de integrale și lista integralelor.
![{\displaystyle \int e^{cx}\;dx={\frac {1}{c}}e^{cx}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6490b98fd68abfbfc525b07948c15652c86eaaac)
![{\displaystyle \int a^{cx}\;dx={\frac {1}{c\ln a}}a^{cx}\qquad {\mbox{(pentru }}a>0,{\mbox{ }}a\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a066f6c8420a9e615034555807a46d4f4364ac7)
![{\displaystyle \int xe^{cx}\;dx={\frac {e^{cx}}{c^{2}}}(cx-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87ff3c663e08f40dd34fa65558256af1342960eb)
![{\displaystyle \int x^{2}e^{cx}\;dx=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/9bf04c5671a90775331e9062a01822af1d47d686)
![{\displaystyle \int x^{n}e^{cx}\;dx={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/418fe2b2ced06b6c35b56ca4fced07c89109d160)
![{\displaystyle \int {\frac {e^{cx}\;dx}{x}}=\ln |x|+\sum _{i=1}^{\infty }{\frac {(cx)^{i}}{i\cdot i!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1677960956614a3f39d2f3ae543d58563c201734)
![{\displaystyle \int {\frac {e^{cx}\;dx}{x^{n}}}={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}}{x^{n-1}}}\,dx\right)\qquad {\mbox{(pentru }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90585ce78d46a3844ab42debdba6dfbb4c60c6eb)
![{\displaystyle \int e^{cx}\ln x\;dx={\frac {1}{c}}e^{cx}\ln |x|-\operatorname {Ei} \,(cx)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33053d64f5ae7f565b07031b02709dc50ee5a109)
![{\displaystyle \int e^{cx}\sin bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b3a37bb249edda5a8d96162f04099cbb2e5720)
![{\displaystyle \int e^{cx}\cos bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3160015214a668d9826f6d1e85dcb7ad045609)
![{\displaystyle \int e^{cx}\sin ^{n}x\;dx={\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\;dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/014136db6b329795471836ce4ac17c86fffbf1cb)
![{\displaystyle \int e^{cx}\cos ^{n}x\;dx={\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\;dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/560942803e654f1b9f1445ce5589a24c3c0ef7ab)
![{\displaystyle \int xe^{cx^{2}}\;dx={\frac {1}{2c}}\;e^{cx^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/220b75633319a42d23352dd64fefbf79e52a4cf9)
![{\displaystyle \int {1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\;dx={\frac {1}{2\sigma }}(1+{\mbox{erf}}\,{\frac {x-\mu }{\sigma {\sqrt {2}}}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1c8c3b8857a3111a5661355182cb664e23c0b0)
- unde
![{\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/54137774455c2d0e732f03ef0f0e060a1fd85ac6)
(Integrala gaussiană)
![{\displaystyle \int _{-\infty }^{\infty }xe^{-a(x-b)^{2}}\,dx=b{\sqrt {\pi \over a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/081fc08d0146fd4822234c1831050486fd76d43e)
![{\displaystyle \int _{-\infty }^{\infty }x^{2}e^{-ax^{2}}\,dx={\frac {1}{2}}{\sqrt {\pi \over a^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f1ea6b45e76cc9a9e7ba24ac8b011e713c034c)
![{\displaystyle \int _{0}^{\infty }x^{2n}e^{-{x^{2}}/{a^{2}}}\,dx={\sqrt {\pi }}{(2n)! \over {n!}}{\left({\frac {a}{2}}\right)}^{2n+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf1a5aece1832b35d62e1645965a8d6b4c33c5fe)
![{\displaystyle \int _{-\infty }^{\infty }e^{-{x^{2}}/{a^{2}}}\cos bx\,dx=a{\sqrt {\pi }}(\sin {a^{2}b^{2} \over 4}+\cos {a^{2}b^{2} \over 4})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7254f2c47724908854aa316dd6939eebf3ab00fd)
(
este funcția Bessel de speța I modificată)
![{\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)
![{\displaystyle \int _{0}^{\infty }x^{a}e^{-bx}dx={\frac {a!}{b^{a+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/774ee1663d71c5a52903a90297ba71ff1bcf30e9)