# 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

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