아래 목록은 지수함수의 적분이다.
부정적분[편집]
각 적분식에서 적분상수
는 생략하였다.
지수함수만 포함하는 함수의 적분[편집]
![{\displaystyle \int f'(x)e^{f(x)}\,dx=e^{f(x)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86148a16ca8ad7f5d0a3e8ce91a6cea111382382)
![{\displaystyle \int e^{cx}\,dx={\frac {1}{c}}e^{cx}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ce7c2dc8ca0393cdbda5681f93f1dd77242177)
![{\displaystyle (a>0,\ a\neq 1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d68811ef1e14db9c175dba97a6c891b42edc525)
다항식을 포함하는 함수의 적분[편집]
![{\displaystyle \int xe^{cx}\,dx=e^{cx}\left({\frac {cx-1}{c^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e64b7471790d705b0da4a5a9022e311661da69ab)
![{\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/ef18e14e28b0e82287b84a971cb3258b25854c8d)
![{\displaystyle {\begin{aligned}\int x^{n}e^{cx}\,dx&={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}\,dx\\&=\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}\\&=e^{cx}\sum _{i=0}^{n}(-1)^{n-i}{\frac {n!}{i!c^{n-i+1}}}x^{i}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2078d7ca28221da215e3e29d932718eac56f0a)
![{\displaystyle \int {\frac {e^{cx}}{x}}\,dx=\ln |x|+\sum _{n=1}^{\infty }{\frac {(cx)^{n}}{n\cdot n!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d72411085d46477bc3ee8fe57d182a2d85c7b854)
![{\displaystyle (n\neq 1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86e7087fd459a0ff4a9caa4624dc87a6049f7e61)
삼각함수를 포함하는 함수의 적분[편집]
(이때
)
(이때
)
![{\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/3050725e6e1d9c9e7020ed33cc62d338ab86583d)
![{\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/fdad87a4bf6155c4e9fb86e3795f79f35626bfc4)
오차함수와 관련된 함수의 적분[편집]
다음 식들에서 erf는 오차 함수이고, Ei는 지수 적분 함수이다.
![{\displaystyle \int e^{cx}\ln x\,dx={\frac {1}{c}}\left(e^{cx}\ln |x|-\operatorname {Ei} (cx)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82f909663662a7e4c1715a44214a26c0ec7ab3b)
![{\displaystyle \int xe^{cx^{2}}\,dx={\frac {1}{2c}}e^{cx^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7581343f0df56ff5a130f1c2c7a3f7b4d7cea832)
![{\displaystyle \int e^{-cx^{2}}\,dx={\sqrt {\frac {\pi }{4c}}}\operatorname {erf} ({\sqrt {c}}x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf6f0ecc13f114c367bed9937a6b9ceffde1a97)
![{\displaystyle \int xe^{-cx^{2}}\,dx=-{\frac {1}{2c}}e^{-cx^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e3ea3b71168d526c08a41b0afa29f2220b21bea)
![{\displaystyle \int {\frac {e^{-x^{2}}}{x^{2}}}\,dx=-{\frac {e^{-x^{2}}}{x}}-{\sqrt {\pi }}\operatorname {erf} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b21fec33291265ce6d5915d45a58b445521390)
![{\displaystyle \int {{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}\,dx={\frac {1}{2}}\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a76f89e34a850c4325348aa1e188c64d769fe9d6)
기타 적분[편집]
- (이때
이고, 모든
에 대해 성립한다.)
- (이때
이고, Γ(x,y)는 불완전 감마 함수이다.)
(이때
,
이고
이다.)
(이때
,
이고
이다.)
![{\displaystyle \int {\frac {ae^{cx}-1}{be^{cx}-1}}\,dx={\frac {(a-b)\log(1-be^{cx})}{bc}}+x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92c83e7c7e4e9291e85042b197cca5e88ef3cada)
정적분[편집]
![{\displaystyle {\begin{aligned}\int _{0}^{1}e^{x\cdot \ln a+(1-x)\cdot \ln b}\,dx&=\int _{0}^{1}\left({\frac {a}{b}}\right)^{x}\cdot b\,dx\\&=\int _{0}^{1}a^{x}\cdot b^{1-x}\,dx\\&={\frac {a-b}{\ln a-\ln b}}(a>0,\ b>0,\ a\neq b)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e42d9a1a6465e1f26a14a77ddd6809ada35ff4cc)
위 적분식의 마지막 값은 로그 평균을 뜻한다.
![{\displaystyle \int _{0}^{\infty }e^{-ax}\,dx={\frac {1}{a}}\quad (\operatorname {Re} (a)>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d17873e603d6115515e3b9697c1a1c7dc6afb5b)
(가우스 적분)
![{\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}\,dx={\sqrt {\pi \over a}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d173c3a0a51d1491a0c0ccb21456ec842d991df1)
![{\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}e^{-{\frac {b}{x^{2}}}}\,dx={\sqrt {\frac {\pi }{a}}}e^{-2{\sqrt {ab}}}\quad (a,b>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72f81ec890d23ed583e6ff3feab019e73c8bf1ec)
![{\displaystyle \int _{-\infty }^{\infty }e^{-(ax^{2}+bx)}\,dx={\sqrt {\pi \over a}}e^{\tfrac {b^{2}}{4a}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3c6c1a2da5557eef70f1e8c104450eb178c0ad)
![{\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}e^{-2bx}\,dx={\sqrt {\frac {\pi }{a}}}e^{\frac {b^{2}}{a}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac314128d031a513d31afafdb437d0545cf0ff62)
![{\displaystyle \int _{-\infty }^{\infty }xe^{-a(x-b)^{2}}\,dx=b{\sqrt {\frac {\pi }{a}}}\quad (\operatorname {Re} (a)>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d04ac61db5ed1f8b014e76248642132670e278)
![{\displaystyle \int _{-\infty }^{\infty }xe^{-ax^{2}+bx}\,dx={\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/8f12419c1c03459f9e4485f00c24d8847701697c)
![{\displaystyle \int _{-\infty }^{\infty }x^{2}e^{-ax^{2}}\,dx={\frac {1}{2}}{\sqrt {\pi \over a^{3}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdc40a04ccfabe052e1faa2b0bc367c3b712a21c)
![{\displaystyle \int _{-\infty }^{\infty }x^{2}e^{-(ax^{2}+bx)}\,dx={\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/e6a4124875e31164bb24d6336c90e53bc52cd5b1)
![{\displaystyle \int _{-\infty }^{\infty }x^{3}e^{-(ax^{2}+bx)}\,dx={\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/568dfd209a063111b62961ad5a6c71329069ca76)
![{\displaystyle \int _{0}^{\infty }x^{n}e^{-ax^{2}}\,dx={\begin{cases}{\dfrac {\Gamma \left({\frac {n+1}{2}}\right)}{2\left(a^{\frac {n+1}{2}}\right)}}&(n>-1,\ a>0)\\\\{\dfrac {(2k-1)!!}{2^{k+1}a^{k}}}{\sqrt {\dfrac {\pi }{a}}}&(n=2k,\ a>0)\\\\{\dfrac {k!}{2(a^{k+1})}}&(n=2k+1,\ a>0)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aefdd640a4d4533905b281f9d5eb21410dc9cc5e)
- (이때
는 정수,
는 이중계승이다.)
![{\displaystyle \int _{0}^{\infty }x^{n}e^{-ax}\,dx={\begin{cases}{\dfrac {\Gamma (n+1)}{a^{n+1}}}&(n>-1,\ \operatorname {Re} (a)>0)\\\\{\dfrac {n!}{a^{n+1}}}&(n=0,1,2,\ldots ,\ \operatorname {Re} (a)>0)\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b4ee138bba46ac2cccac62668472c45bbcab3ce)
![{\displaystyle \int _{0}^{1}x^{n}e^{-ax}\,dx={\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/92482c2e5d7502755c6da9b6f088ff00721580e1)
![{\displaystyle \int _{0}^{b}x^{n}e^{-ax}\,dx={\frac {n!}{a^{n+1}}}\left[1-e^{-ab}\sum _{i=0}^{n}{\frac {(ab)^{i}}{i!}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f17026bf5a6616142c6b1a8f0392f3ceb373cbbd)
![{\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/520ba3106679a3134d097708f1920c93a7fa51da)
![{\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/2b472dd31344bfdeeb1d6b1129996af0083b324c)
![{\displaystyle \int _{0}^{\infty }e^{-ax}\sin bx\,dx={\frac {b}{a^{2}+b^{2}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10f8d0c56030576a9ea5c32fdcc26da56cf84bc7)
![{\displaystyle \int _{0}^{\infty }e^{-ax}\cos bx\,dx={\frac {a}{a^{2}+b^{2}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ecb94f832c0bd6d7d22c0d1a6a0a2d05f982f9)
![{\displaystyle \int _{0}^{\infty }xe^{-ax}\sin bx\,dx={\frac {2ab}{(a^{2}+b^{2})^{2}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4364f14319b127a45b9e81e92c7777ba6a850e2a)
![{\displaystyle \int _{0}^{\infty }xe^{-ax}\cos bx\,dx={\frac {a^{2}-b^{2}}{(a^{2}+b^{2})^{2}}}\quad (a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d93ccc76421b3d124dcedd0972c92a9769063658)
![{\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}\sin bx}{x}}\,dx=\arctan {\frac {b}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2db30c7b2a9469e9a5bfcbcf1e89ff008e0962)
![{\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}-e^{-bx}}{x}}\,dx=\ln {\frac {b}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2175f6ab38d3bcbda8248ff8539181b3374e4aed)
![{\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}-e^{-bx}}{x}}\sin px\,dx=\arctan {\frac {b}{p}}-\arctan {\frac {a}{p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7649784af743a103173c755ff751b781f28a0707)
![{\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}-e^{-bx}}{x}}\cos px\,dx={\frac {1}{2}}\ln {\frac {b^{2}+p^{2}}{a^{2}+p^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34ac535750fd596ab737ac5fcdc48fa925ac6842)
![{\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}(1-\cos x)}{x^{2}}}\,dx=\operatorname {arccot} a-{\frac {a}{2}}\ln(a^{2}+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a033cf3a2160c224e40b7658bd77e0a5bec45d04)
(I0는 제1종 변형 베셀 함수이다.)
![{\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)
(
는 다중로그이다.)
![{\displaystyle \int _{0}^{\infty }{\frac {\sin mx}{e^{2\pi x}-1}}\,dx={\frac {1}{4}}\coth {\frac {m}{2}}-{\frac {1}{2m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d62701d01afce173f25bba4aab3710da2c2eadf)
(
는 오일러-마스케로니 상수)
같이 보기[편집]