Indefinite integral

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, $C$, is always added.

A list of simple antiderivatives

The identity antiderivative:

$\int dx=x+C$

Polynomial and simple rational

$\int xdx={1 \over 2}x^{2}+C$

$\int x^{2}dx={1 \over 3}x^{3}+C$

$\int {1 \over {x^{3}}}dx=\int x^{-3}dx={1 \over -3}x^{-2}+C={-1 \over {3x^{-2}}}+C$

$\int x^{3.873}dx={1 \over 4.873}x^{4.873}+C$

The general rule for polynomial expressions is:

$\int x^{n}dx={1 \over {n+1}}x^{n+1}+C$

Note: $n\neq -1$. See below for when $n=-1$

Rational

For a more detailed treatment, see Partial fractions in integration.
Rational antiderivatives are much more difficult and follow different rules.

$\int {1 \over x}dx=\int x^{-1}dx=\ln(x)+C$

$\int {1 \over {x+1}}dx=\ln(x+1)+C$

$\int {1 \over {x+a}}dx=\ln(x+a)+C$

Trigonometric

$\int \sin(x)dx=-\cos(x)+C$

$\int \cos(x)dx=\sin(x)+C$

$\int \tan(x)dx=\ln |\sec(x)|+C$

$\int \sin(x)dx=-\cos(x)+C$

Exponential

$\int e^{x}=e^{x}+C$

$\int a^{x}=\ln(a)a^{x}+C$

See also

• Integral
• Definite integral