# Antiderivative

How to find the antiderivative of a given function: definition, examples, and their solutions.

## Definition

Integration is finding the antiderivative:

the opposite of finding the derivative.

The antiderivative of *F*'(*x*) is *F*(*x*) + *C*.

(*C*, the constant of integration, is added,

because no matter what constant *C* is,

that *C* always satisfies [*F*(*x*) + *C*]' = *F*'(*x*).)

The antiderivative of *f*(*x*) is '∫ *f*(*x*) *dx*'.

('∫ *f*(*x*) *dx*' is read as 'the integral of *f*(*x*) d.x.'.)

So, to find ∫ *f*(*x*) *dx*,

find the antiderivative of *f*(*x*): *F*(*x*) + *C*.

## Example 1

Think of the antiderivative of 3*x*^{2}.

[*x*^{3}]' = 3*x*^{2}

So the antiderivative is *x*^{3}.

Power rule in differentiation (Part 1)

So (given) = *x*^{3} + *C*.

Don't forget to add +*C*.

## Example 2

Think of the antiderivative of 5*x*^{4}.

[*x*^{5}]' = 5*x*^{4}

So the antiderivative is *x*^{5}.

Power rule in differentiation (Part 1)

So (given) = *x*^{5} + *C*.

## Example 3

∫ *dx* means ∫ 1 *dx*.

So think of the antiderivative of 1.

[*x*]' = 1

So the antiderivative is *x*.

Power rule in differentiation (Part 1)

So (given) = *x* + *C*.