Substitution Rule

Basic Problems

Example (1)

Example (2)

Example (3)

Example (4)

Example (5)

Example (6)

Substitution Rule Definite Integral Case

Example (1)

Example (2)

Example (3)

Substitution Rule More Challenging Problems

Example (1)

Method 1

Note that the first method can be used to find the integral of any function of the form: f(x) = x (2n-1) (ax n +b) k for any positive integer n and any real number k (where k is not -1) as the following examples show:

Example (2)

In all of the first three examples, we let: u = 2x+ 4 and so: du = 2dx → dx = du/2 and x = (u - 4)/2

In the fourth example, we let: u = 2x and so: du = 4xdx → dx = du/4x and x 2 = (u - 4)/2

In the fifth example, we let: u = 2x and so: du = 6x 2 dx → dx = du/6x 2 and x 3 = (u - 4)/2

Examples (3)

The double angle formulas can simplify these problems, by replacing cos 2 x by (1+cos2x)/2 and sin 2 x by (1- cos2x)/2

Note: If the problems were what we have below, then his would be like the basic examples. Do them!