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

Answers to Above Exercises

1. - (1/5)sin ⁵ (x) + (1/3)sin ³ (x)
2. (1/17)cos ¹⁷ (x) - (1/15)cos ¹⁵ (x)

More references and Links

integrals and their applications in calculus.