Differentiation asks for the rate of change of a function. An antiderivative reverses that process.

A function $F$ is an antiderivative of $f$ on an interval when

$$F'(x)=f(x).$$

For example, because

$$\frac{d}{dx}x^3=3x^2,$$

one antiderivative of $3x^2$ is $x^3$.

The constant of integration

Derivatives do not detect added constants:

$$\frac{d}{dx}(x^3+C)=3x^2.$$

Therefore all antiderivatives of the same function on an interval differ by a constant. This is written

$$\int f(x),dx=F(x)+C,$$

where the integral sign without bounds denotes an indefinite integral.

Reversing derivative rules

Basic antiderivatives come from reading derivative formulas backwards. For example,

$$\int x^n,dx=\frac{x^{n+1}}{n+1}+C\qquad(n\ne-1),$$

$$\int \frac1x,dx=\ln|x|+C$$

on intervals not containing zero, and

$$\int \cos x,dx=\sin x+C.$$

Finding antiderivatives is different from defining a definite integral as an accumulation. The fundamental theorem of calculus explains why these two ideas are connected.