Tutorial to find integrals involving the product of powers of sin(x) and cos(x) with one of the two having an odd power. Examples and exercises with solutions are included.

In what follows, C is the constant of integration.

The main idea is to rewrite the integral writing the term with the odd power as the product of a term with power 1 and a term with an even power. Example: sin

sin

We now use the identity sin

sin

We now let u = cos(x), hence du/dx = -sin(x) or -du = sin(x)dx and substitute in the given integral to obtain

sin

Expand and calculate the integral on the right

sin

= (1/5)u

Substitute u by cos(x) to obtain

sin

Rewrite cos

sin

We now use the identity cos

sin

We now let u = sin(x), hence du/dx = cos(x) or du = cos(x)dx and substitute in the given integral to obtain

sin

Expand and calculate the integral on the right

sin

= (u

= (1/17)u

Substitute u by sin(x) to obtain

sin

1. cos

2. sin

2. (1/17)cos