Indefinite integral
| <math>\frac{d}{dx} \sin x=?\,</math> | This article/section deals with mathematical concepts appropriate for late high school or early college. |
An indefinite integral, or antiderivative, is an integral without upper and lower limits.
Indefinite Integrals
There are an infinite number of antiderivatives for a given function, because each indefinite integral can have an arbitrary constant added to it which disappears upon differentiation. However, the fundamental theorem of calculus relates a definite integral to an indefinite integral by taking its value at the boundary points.
Whenever any expression is integrated the constant of integration, <math>C</math>, is always added.
A list of simple antiderivatives
The identity antiderivative:
- <math>\int dx = x + C</math>
Polynomial and simple rational
To see the proofs for the first two integrals, see Riemann integral.
- <math>\int xdx = {1 \over 2} x^2 + C</math>
- <math>\int x^2dx = {1 \over 3} x^3 + C</math>
- <math>\int {1 \over {x^3}}dx = \int x^{-3}dx = {1 \over -3}x^{-2} + C = {-1 \over {3x^{-2}}} + C</math>
- <math>\int x^{3.873}dx = {1 \over 4.873} x^{4.873} + C</math>
The general rule for polynomial expressions is:
- <math>\int x^ndx = {1 \over {n+1}} x ^ {n + 1} + C</math>
Note: <math>n \ne -1</math>. See below for when <math>n=-1</math>
Rational
For a more detailed treatment, see Partial fractions in integration.
Rational antiderivatives are much more difficult and follow different rules.
- <math>\int {1 \over x}dx = \int x^{-1}dx = \ln(x) + C</math>
- <math>\int {1 \over {x+1}}dx = \ln(x+1) + C</math>
- <math>\int {1 \over {x+a}}dx = \ln(x+a) + C</math>
Trigonometric
- <math>\int \sin(x)dx = -\cos(x) + C</math>
- <math>\int \cos(x)dx = \sin(x) + C</math>
- <math>\int \tan(x)dx = \ln|\sec(x)| + C</math>
Exponential
- <math>\int e^x = e^x + C</math>
- <math>\int a^x = {a^x \over \ln(a)} + C</math>