적분은 미적분학의 두 기본연산 중의 하나이다. 적분은 미분처럼 복잡한 함수를 보다 간단한 함수들로 분해하여 계산할 수는 없기 때문에, 여러 함수에 대한 적분을 모아 놓은 적분표는 유용하게 사용된다.
아래의 식들에서 C는 적분 상수이다.
일반적인 적분 규칙[편집]
![{\displaystyle \int af(x)\,dx=a\int f(x)\,dx\qquad {\mbox{(}}a{\mbox{ constant)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f65f80a8ad928786221fa055b2a5e2d651f0a1ab)
![{\displaystyle \int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc2c6e2acadf432d22ca42bc6a21af25e48e64d)
![{\displaystyle \int f(x)g(x)\,dx=f(x)\int g(x)\,dx-\int \left[f'(x)\left(\int g(x)\,dx\right)\right]\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/054fd882152acf3191c73c0a0eda7256b2ecac74)
![{\displaystyle \int [f(x)]^{n}f'(x)\,dx={[f(x)]^{n+1} \over n+1}+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f01c72f41d171a9b2ff4cb3f77589b1affc2a98)
![{\displaystyle \int {f'(x) \over f(x)}\,dx=\ln {\left|f(x)\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d00d43661f34a6a78da766e3e92b83b311b51a45)
![{\displaystyle \int {f'(x)f(x)}\,dx={1 \over 2}[f(x)]^{2}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55ddd18a718860218bb3d10a0ccc72af8e420c58)
적분표[편집]
아래 문서들에서 다양한 적분 공식들을 찾아볼 수 있다.
간단한 함수의 적분[편집]
![{\displaystyle \int a\,dx=ax+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47cda1f818c870d8c4f987c56eeaea4095a5a47e)
![{\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ if }}n\neq -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a59db83321801fcc1ef66f06a2da5088ba746765)
![{\displaystyle \int (ax+b)^{n}\,dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad {\text{(for }}n\neq -1{\text{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1f87606f321330ae1c1d56a49e39730b0dd0ea)
![{\displaystyle \int {1 \over x}\,dx=\ln {\left|x\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5381eb598bc2fa34b8ef2c9b186fb5afbc3f8ffb)
![{\displaystyle \int {\frac {c}{ax+b}}\,dx={\frac {c}{a}}\ln \left|ax+b\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a2b702fce460d1ad3899bcac1720aab5f9f6406)
![{\displaystyle \int {\frac {1}{1+x^{2}}}\,dx=\arctan {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3042a46df402bbedb1798be605d923370ca2c4)
무리함수[편집]
![{\displaystyle \int {1 \over {\sqrt {1-x^{2}}}}\,dx=\arcsin {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/110be345f65b6d301ef99add95df008aecbac43b)
![{\displaystyle \int {-1 \over {\sqrt {1-x^{2}}}}\,dx=\arccos {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5324315292bd8b5c3eeb0e39a6a230a802ac131)
![{\displaystyle \int {1 \over |x|{\sqrt {x^{2}-1}}}\,dx={\mbox{arcsec}}\,{x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8468b2cfd90cf3e115802a19b4af9e41a705e43e)
![{\displaystyle \int \ln {x}\,dx=x\ln {x}-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3987107bc91f514f684ac40e88852dc47e33abc9)
![{\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bcd50506a3f83a358205e51f289e179c84b4d94)
![{\displaystyle \int e^{x}\,dx=e^{x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6d31e8ad38cc40b4e3d18ad17b756efa483abd)
![{\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/816dde2034e43093b2a85e3dcc1ef2f39779f860)
![{\displaystyle \int \cos {x}\,dx=\sin {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1aae2ec756513ea8f93deb874803c61e291dd8a)
![{\displaystyle \int \sin {x}\,dx=-\cos {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/537de256cbb401203900fd3623cdbc85e31cc70b)
![{\displaystyle \int \tan {x}\,dx=-\ln {\left|\cos {x}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db65697f06c5abbe10e8ff7dbf78d1213439495f)
![{\displaystyle \int \csc {x}\,dx=\ln {\left|\csc {x}-\cot {x}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/912b48f413446f1ea54ceeec71a2f7a4f6808e42)
![{\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/378b45f5cd66c9fb7560eb362481df12ce77fa51)
![{\displaystyle \int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a79422c3c1bc1b58e8a1623920b50fb4ff87f907)
![{\displaystyle \int \sec ^{2}x\,dx=\tan x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f8fbfacf62d7130b7bf000e226b07f8c599bf1c)
![{\displaystyle \int \csc ^{2}x\,dx=-\cot x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/364c3afec409bb6bfbb787276d7cfd884040b07a)
![{\displaystyle \int \sin ^{2}mx\,dx={{\frac {1}{2m}}(mx-\sin mx\cos mx)}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a82656ca0c3753bceca89f005debd9387660de0)
![{\displaystyle \int \cos ^{2}mx\,dx={{\frac {1}{2m}}(mx+\sin mx\cos mx)}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d3e6835f20c0bfc4072150ed22550565c296c91)
![{\displaystyle \int \sin ^{n}x\,dx={-{\frac {\sin ^{n-1}x\cos x}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}x\,dx}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e23f8f24b53432fcb3f8cd0f8b3cf0b3ec4ac2e8)
![{\displaystyle \int \cos ^{n}x\,dx={{\frac {\cos ^{n-1}x\sin x}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}x\,dx}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2be0e4fce515dc377246c7a12605950f40eb841)
![{\displaystyle \int \sec ^{n}x\,dx={{\frac {\sec ^{n-2}x\tan x}{n-1}}+{\frac {n-2}{n-1}}\int \sec ^{n-2}x\,dx}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd6ab3588f1e17839be46644cca99ad4a0827a9)
![{\displaystyle \int \csc ^{n}x\,dx={{\frac {\csc ^{n-2}x\cot x}{-(n-1)}}+{\frac {n-2}{n-1}}\int \csc ^{n-2}x\,dx}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1706b0598ea5323b24f3b2c7feccbb44ad6d3c70)
![{\displaystyle \int \sinh x\,dx=\cosh x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a452d5b48cae9335f0a79d19b85a61d28154683a)
![{\displaystyle \int \cosh x\,dx=\sinh x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/529344aa89d4a7732c58734fa5134612b73aaa19)
![{\displaystyle \int \tanh x\,dx=\ln(\cosh x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3d5e34f541525e5ce502d308796904a0dfba1b)
![{\displaystyle \int {\mbox{csch}}\,x\,dx=\ln \left|\tanh {x \over 2}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53e784b82203b2db8d3bb9435d677aa204705ef1)
![{\displaystyle \int {\mbox{sech}}\,x\,dx=\arctan(\sinh x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cba1e31fd35b44ba1cd78d5ec48f68be1d5f7a8)
![{\displaystyle \int \coth x\,dx=\ln |\sinh x|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd7bd1bfe08e160d8d488e245bd13c42a16c91d)
정적분[편집]
어떤 함수의 적분은 원시 함수로 나타낼 수 없지만, 특정 구간에서의 적분값을 계산할 수는 있다. 다음은 그들 중 유용한 몇 정적분이다.
![{\displaystyle \int _{0}^{\infty }{{\sqrt {x}}e^{-x}\,dx}={\frac {1}{2}}{\sqrt {\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/530dfc58a96deef4dccee6aed406f8b3a3c2d2f7)
![{\displaystyle \int _{0}^{\infty }{e^{-x^{2}}\,dx}={\frac {1}{2}}{\sqrt {\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16b05b4bc20fb5de48a721eed507e9f61580d0b3)
외부 링크[편집]