Tässä on luettelo tavallisimpien matemaattisten operaattoreiden sekä funktioiden integraaleista.
Rationaalifunktiot
![{\displaystyle \int k\,dx=kx+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39fba8999cf049c9826f334d9ee25938865aeb71)
![{\displaystyle \int x^{a}\,dx={\frac {x^{a+1}}{a+1}}+C\qquad \ (a\neq -1{\text{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e329a912df13984a28ee3fedb725e1fa0df458a7)
![{\displaystyle \int (ax+b)^{n}\,dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad \ (n\neq -1{\text{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c78a42d2186c102e00aae599c69def7f20c6f9d)
![{\displaystyle \int {1 \over x}\,dx=\ln \left|x\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e45d8b935713d6a90372cb1d83f9f8c37732b21)
![{\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)
Eksponenttifunktiot
![{\displaystyle \int e^{ax}\,dx={\frac {1}{a}}e^{ax}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c41b04a88938a3a50de7ca4af3c567b33f9ccb37)
![{\displaystyle \int f'(x)e^{f(x)}\,dx=e^{f(x)}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91a811704bafdfdebab7ccdc39f6dedca25a1bfd)
![{\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/544302360e06c836a24783781373750b9cb709d6)
Logaritmifunktiot
![{\displaystyle \int \ln x\,dx=x\ln x-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54b730b8defb012d4140543f799c9fb1a7f1a3c8)
![{\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)
Trigonometriset funktiot
![{\displaystyle \int \sin {x}\,dx=-\cos {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/537de256cbb401203900fd3623cdbc85e31cc70b)
![{\displaystyle \int \cos {x}\,dx=\sin {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1aae2ec756513ea8f93deb874803c61e291dd8a)
![{\displaystyle \int \tan {x}\,dx=-\ln {\left|\cos {x}\right|}+C=\ln {\left|\sec {x}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fff79bf847dd0567db9119faab36c03810a868c8)
![{\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 {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/378b45f5cd66c9fb7560eb362481df12ce77fa51)
![{\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 ^{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 \sec {x}\,\tan {x}\,dx=\sec {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/385d180bf75e276f8b0cafb1fdc1f584554be54f)
![{\displaystyle \int \csc {x}\,\cot {x}\,dx=-\csc {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/038c3e132b5c6826b7be055d24fa617842c493d2)
![{\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}}\left(x-{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x-\sin x\cos x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3915367234cede7f4f2606aacaf32b35cfcf3e)
![{\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}\left(x+{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x+\sin x\cos x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c5fd33e2ed813a3b2452c5b7bc553991b1855ea)
![{\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e685542d54058f2defcf20dad355de10535d8d5)
![{\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c15d560c4a9f07da5aa62b1adc435b6e785ea33)
![{\displaystyle \int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/307e19c74642ddbed625e25265cb0ee59638d286)
Arkusfunktiot
![{\displaystyle \int \arcsin {x}\,dx=x\arcsin {x}+{\sqrt {1-x^{2}}}+C,\vert x\vert \leq +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/219c6f13f7a40457004d8c34c039563acbca7648)
![{\displaystyle \int \arccos {x}\,dx=x\arccos {x}-{\sqrt {1-x^{2}}}+C,\vert x\vert \leq +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef65dc20d5f13f08dd64786845c8d49253661ab4)
![{\displaystyle \int \arctan {x}\,dx=x\arctan {x}-{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebfcbb8c9599fc72ae67877592dceed0d191ae8b)
![{\displaystyle \int \operatorname {arccot} {x}\,dx=x\operatorname {arccot} {x}+{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc06b6a5a0a63e699fd2409c29a4c88847adad35)
![{\displaystyle \int \operatorname {arcsec} {x}\,dx=x\operatorname {arcsec} {x}-\ln \vert x\,(1+{\sqrt {1-x^{-2}}}\,)\vert +C,\vert x\vert \geq +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e646eabda8dccd22b6f9d82fe40ee0639f477cde)
![{\displaystyle \int \operatorname {arccsc} {x}\,dx=x\operatorname {arccsc} {x}+\ln \vert x\,(1+{\sqrt {1-x^{-2}}}\,)\vert +C,\vert x\vert \geq +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d8ed817c64c2381eb85df2f60d63269bc6fd15)
Hyperboliset funktiot
![{\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/d34084777ab4d5122b6fb3a917d140df1caad0dd)
![{\displaystyle \int \coth x\,dx=\ln |\sinh x|+C,x\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b28e78ca3fc07cd577a9935aa4b697eb393d65da)
![{\displaystyle \int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c567185304799602087bcbe1b470a2b9e5b7880b)
![{\displaystyle \int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C,x\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df142652b95d4d0ba392882a24d561bcaf825d60)
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2}}\right|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d14b674a9b740e8c1fe8b3a534365953d3993dd4)
![{\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {2}{a}}\arctan e^{ax}+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0dff59f32e29e40ac43a4f357101cdcc9792acf)
Lähteet
- Valtanen, Esko: Matemaattisia kaavoja ja taulukoita, s. 78–80. , 2013. ISBN 978-952-9867-37-0.
Kirjallisuutta
- Spiegel, Murray M.: Mathematical Handbook of Formulas and Tables. Shaum's Outline Series. McGraw-Hill, 1992. ISBN 0-07-060224-7.