Indefinite integral

 $\frac{d}{dx} \sin x=?\,$ This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

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

To see the proofs for the first two integrals, see Riemann integral.

$\int xdx = {1 \over 2} x^2 + C$
$\int x^2dx = {1 \over 3} x^3 + C$
$\int {1 \over {x^3}}dx = \int x^{-3}dx = {1 \over -3}x^{-2} + C = {-1 \over {3x^{-2}}} + C$
$\int x^{3.873}dx = {1 \over 4.873} x^{4.873} + C$

The general rule for polynomial expressions is:

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

Note: $n \ne -1$. 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

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