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<center><math>\int \ln |x| \, dx = x \ln |x| - x</math>.</center>
=== Integroimiskaavoja ===
=== Yleinen luettelo integroimiskaavoista ===
Alla olevissa kaavoissa <math>f</math> ja <math>g</math> ovat <math>x</math>:stä riippuvia Riemann-integroituvia funktioita, <math>k</math> reaaliluku ja <math>a</math> positiivinen reaaliluku.
{| align="center"
|align=right|:<math>\int kf \ sin, dx = k \int xf \, dx</math> ▼|-
|align=right|:<math>\int kff'g \, dx = fg - \int g'f \, dx</math>
|align="center"|:<math>\int (f+g) \, dx = \int f \, dx + \int g \, dx</math>
|align=left|:<math>k \int f0 \, dx = k</math>
|align=right|:<math>\int k \, dx = kx</math> ▼|-
|align=right|:<math>\int f'gx^n \, dx = \frac{1}{n+1} x^{n+1} \, , \ \mathrm{kun} \ n \ne -1</math>
|align="center"|:<math>\int \frac{1}{x} \, dx = \ln |x|\,</math>
|align=left|:<math>fg - \int gf'f^n \, dx = \frac{1}{n+1} f^{n+1} \, , \ \mathrm{kun} \ n \ne -1</math>
|align=right|:<math>\int \frac{f'}{f} \, dx = \ln |f|</math> ▼|-
|align=right|:<math>\int (f+g)\sin x \, dx = - \cos x</math>
|align="center"|:<math>\int \cos x \, dx = \sin x\,</math>
|align=left|:<math>\int f\tan x \, dx += -\intln g|\cos x|\, dx</math>
|align=right|:<math>\int f' e^f \, dx = e^f</math> ▼|-
|align=right|:<math>\int 0a^x \, dx = \frac{a^x}{\ln a}</math>
|align="center"|:<math>=\int \ln |x| \, dx = x \ln |x| - x</math>
|align=left|:<math> \int \log_a x \, dx = \log_a e (x \ln |x| - x) \, </math> , \ \mathrm{kun } \ <math>a > 0 \ ,</math> \mathrm{ja } \ <math>a \ne 1</math> ▼|align=left|<math>0\,</math>
|-
▲|align=right|<math>\int k \, dx</math>
|align="center"|<math>=\,</math>
|align=left|<math>kx\,</math>
|-
|align=right|<math>\int x^n \, dx</math>
|align="center"|<math>=\,</math>
|align=left|<math>\frac{1}{n+1} x^{n+1}</math>, kun <math>n \ne -1</math>
|-
|align=right|<math>\int \frac{1}{x} \, dx</math>
|align="center"|<math>=\,</math>
|align=left|<math>\ln |x|\,</math>
|-
|align=right|<math>\int f'f^n \, dx</math>
|align="center"|<math>=\,</math>
|align=left|<math>\frac{1}{n+1} f^{n+1}</math>, kun <math>n \ne -1</math>
|-
▲|align=right|<math>\int \frac{f'}{f} \, dx</math>
|align="center"|<math>=\,</math>
|align=left|<math>\ln |f|\,</math>
|-
▲|align=right|<math>\int \sin x \, dx</math>
|align="center"|<math>=\,</math>
|align=left|<math>-\cos x\,</math>
|-
|align=right|<math>\int \cos x \, dx</math>
|align="center"|<math>=\,</math>
|align=left|<math>\sin x\,</math>
|-
|align=right|<math>\int \tan x \, dx</math>
|align="center"|<math>=\,</math>
|align=left|<math>-\ln |\cos x|\,</math>
|-
▲|align=right|<math>\int f' e^f \, dx</math>
|align="center"|<math>=\,</math>
|align=left|<math>e^f\,</math>
|-
|align=right|<math>\int a^x \, dx</math>
|align="center"|<math>=\,</math>
|align=left|<math>\frac{a^x}{\ln a}</math>
|-
|align=right|<math>\int \ln |x| \, dx</math>
|align="center"|<math>=\,</math>
|align=left|<math>x \ln |x| - x\,</math>
|-
|align=right|<math>\int \log_a x \, dx</math>
|align="center"|<math>=\,</math>
▲|align=left|<math>\log_a e (x \ln |x| - x)\,</math>, kun <math>a > 0\,</math> ja <math>a \ne 1</math>
|}
== Aiheesta muualla ==
| 