# Integrals Involving sin(x) and cos(x) with Odd Power

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.

## Examples with Detailed solutions

In what follows, C is the constant of integration.

### Example 1

Evaluate the integral sin3(x) cos2(x) dx

Solution to Example 1:
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: sin3(x) = sin2(x) sin(x). Hence the given integral may be written as follows: sin 3 (x) cos 2 (x) dx = sin 2 (x) cos 2 (x) sin(x) dx
We now use the identity sin2(x) = 1 - cos2(x) and rewrite the given integral as follows: sin 3 (x) cos 2 (x) dx = (1 - cos 2 (x)) cos 2 (x) sin(x) dx

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 3 (x) cos 2 (x) dx = - (1 - u 2 ) u 2 du

Expand and calculate the integral on the right sin 3 (x) cos 2 (x) dx = u 4 - u 2 du
= (1/5)u
5 - (1/3)u 3 + C
Substitute u by cos(x) to obtain sin 3 (x) cos 2 (x) dx = (1/5)cos 5 (x) - (1/3)cos 3 (x) + C

### Example 2

Evaluate the integral sin12(x) cos5(x) dx

Solution to Example 2:
Rewrite cos5(x) as follows cos5(x) = cos4(x) cos(x). Hence the given integral may be written as follows: sin 12 (x) cos 5 (x) dx = sin 12 (x) cos 4 (x) cos(x) dx

We now use the identity cos2(x) = 1 - sin2(x) to rewrite cos4(x) in terms of power of sin(x) and rewrite the given integral as follows: sin 12 (x) cos 5 (x) dx = sin 12 (x) (1 - sin 2 (x)) 2 cos(x) dx

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 12 (x) cos 5 (x) dx = u 12 (1 - u 2 ) 2 du

Expand and calculate the integral on the right sin 12 (x) cos 5 (x) dx = u 12 (1 + u 4 - 2u 2 ) du
= (u 16 - 2u 14 + u 12 ) du
= (1/17)u
17 - (2/15)u 15 + (1/13)u 13 + C
Substitute u by sin(x) to obtain sin 3 (x) cos 2 (x) dx = (1/17)sin 17 (x) - (2/15)sin 15 (x) + (1/13)sin 13 (x) + C

## Exercises

Evaluate the following integrals.
1. cos 3 (x) sin 2 (x) dx
2. sin 3 (x) cos 14 (x) dx