Antiderivative

Antiderivative

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

Definition

The antiderivative of F'(x) is F(x) + C. If f(x) = F'(x), then the antiderivative of f(x) is 'the integral of f(x) dx', which is F(x) + C. C is the constant of integration.

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

Find the given indefinite integral. The integral of 3x^2 dx

Think of the antiderivative of 3x2.

[x3]' = 3x2
So the antiderivative is x3.

Power rule in differentiation (Part 1)

So (given) = x3 + C.

Don't forget to add +C.

Example 2

Find the given indefinite integral. The integral of 5x^4 dx

Think of the antiderivative of 5x4.

[x5]' = 5x4
So the antiderivative is x5.

Power rule in differentiation (Part 1)

So (given) = x5 + C.

Example 3

Find the given indefinite integral. The integral of dx

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.