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Integration is one of the two basic operations in calculus and since it, unlike differentiation, is non-trivial, tables of known integrals are often useful.
This page lists some of the most common antiderivatives; a more complete list can be found in the List of integrals.
We use C for an arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinitude of antiderivatives.
These formulas only state in another form the assertions in the table of derivatives.
Rules for integration of general functions
- <math>\int cf(x)\,dx = c\int f(x)\,dx<math>
- <math>\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx<math>
- <math>\int f(x)g(x)\,dx = f(x)\int g(x)\,dx - \int \left(d[f(x)]\int g(x)\,dx\right)<math>
Integrals of simple functions
- more integrals: List of integrals of rational functions
- <math>\int \,dx = x + C<math>
- <math>\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\qquad\mbox{ if }n \ne -1<math>
- <math>\int \frac{1}{x}\,dx = \ln{\left|x\right|} + C<math>
- <math>\int \frac{1}{1+x^2} \, dx = \arctan{x} + C<math>
- more integrals: List of integrals of logarithmic functions
- <math>\int \ln {x}\,dx = x \ln {x} - x + C<math>
- <math>\int \log_b {x}\,dx = x\log_b {x} - x\log_b {e} + C<math>
- more integrals: List of integrals of exponential functions
- <math>\int e^x\,dx = e^x + C<math>
- <math>\int a^x\,dx = \frac{a^x}{\ln{a}} + C<math>
- more integrals: List of integrals of irrational functions
- <math>\int {1 \over \sqrt{1-x^2}} \, dx = \arcsin {x} + C<math>
- <math>\int {-1 \over \sqrt{1-x^2}} \, dx = \arccos {x} + C<math>
- <math>\int {x \over \sqrt{x^2-1}} \, dx = \mbox{arcsec}\,{x} + C<math>
- more integrals: List of integrals of trigonometric functions and List of integrals of arc functions
- <math>\int \sin{x}\, dx = -\cos{x} + C<math>
- <math>\int \cos{x}\, dx = \sin{x} + C<math>
- <math>\int \tan{x} \, dx = -\ln{\left| \cos {x} \right|} + C<math>
- <math>\int \cot{x} \, dx = \ln{\left| \sin{x} \right|} + C<math>
- <math>\int \sec{x} \, dx = \ln{\left| \sec{x} + \tan{x}\right|} + C<math>
- <math>\int \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + C<math>
- <math>\int \sec^2 x \, dx = \tan x + C<math>
- <math>\int \csc^2 x \, dx = -\cot x + C<math>
- <math>\int \sec{x} \, \tan{x} \, dx = \sec{x} + C<math>
- <math>\int \csc{x} \, \cot{x} \, dx = - \csc{x} + C<math>
- <math>\int \sin^2 x \, dx = \frac{1}{2}(x - \sin x \cos x) + C<math>
- <math>\int \cos^2 x \, dx = \frac{1}{2}(x + \sin x \cos x) + C<math>
- <math>\int \arctan{x} \, dx = x \, \arctan{x} + \ln \sqrt{1 + x^2} + C<math>
- more integrals: List of integrals of hyperbolic functions
- <math>\int \sinh x \, dx = \cosh x + C<math>
- <math>\int \cosh x \, dx = \sinh x + C<math>
- <math>\int \tanh x \, dx = \ln (\cosh x) + C<math>
- <math>\int \mbox{csch}\,x \, dx = \ln\left| \tanh {x \over2}\right| + C<math>
- <math>\int \mbox{sech}\,x \, dx = \arctan(\sinh x) + C<math>
- <math>\int \coth x \, dx = \ln|\sinh x| + C<math>
Definite integrals
There are some functions whose antiderivatives cannot be expressed in closed form. However, the values of the definite integrals of some of these functions over some common intervals can be calculated. A few useful definite integrals are given below.
- <math>\int_0^\infty{\sqrt{x}\,e^{-x}\,dx} = \frac{1}{2}\sqrt \pi<math>
- <math>\int_0^\infty{e^{-x^2}\,dx} = \frac{1}{2}\sqrt \pi<math>
- <math>\int_0^\infty{\frac{x}{e^x-1}\,dx} = \frac{\pi^2}{6}<math>
- <math>\int_0^\infty{\frac{x^3}{e^x-1}\,dx} = \frac{\pi^4}{15}<math>
- <math>\int_0^\infty\frac{\sin(x)}{x}\,dx=\frac{\pi}{2}<math>
- <math>\int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z)<math>
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