In this section, we introduce the idea of the indefinite integral. We also look at the process of variable substitution to find antiderivatives of more complex functions.

Section 5.4 Finding Antiderivatives: Substitution

Definition

For any function f, is called the indefinite

integral of f and represents the most general

antiderivative of f.

Definition

For any function f, is called the indefinite

integral of f and represents the most general

antiderivative of f.

For example:

Example 1

Find each of the following:

(a)

(b)

(c)

Example 1cont.

Find each of the following:

(d)

(e)

(f)

Substitution

What if the integrand is not something that we recognize as a “basic” antiderivative rule?

For example,

We would like to do a variable substitution so that the “new” integral is one that we recognize as a “basic” antiderivative rule.

TheoremChange of Variables in an

Integral

Let f, u, and g be continuous functions such that:

for all x.

Then:

The Process

Substitute: Choose a function u = u(x) such that the substitution of u for x and du for dx changes into

Antidifferentiate: Solve - that is, find G(u) such that

Resubstitute: Substitute x back in for u to get the answer to have an antiderivative of the original function

Our Example

Looking again at

Example 2

Find each of the following:

(a)

(b)

Example 3

Find each of the following:

(a)

(b)

Example 4

Find each of the following:

(a)

(b)

TheoremChange of Variables in a Definite

Integral

Let f, u, and g be continuous functions such that:

for all x.

Then:

Example 5

Find each of the following:

(a)

(b)