# 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. Exercises with answers are at the bottom of the page.

## Examples with Detailed solutionsIn what follows, C is the constant of integration.
## Example 1Evaluate the integral^{3}(x) cos^{2}(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: sin ^{3}(x) = sin^{2}(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 sin ^{2}(x) = 1 - cos^{2}(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 intergral 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 2Evaluate the integral^{12}(x) cos^{5}(x) dx
Solution to Example 2:Rewrite cos ^{5}(x) as follows cos^{5}(x) = cos^{4}(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 cos ^{2}(x) = 1 - sin^{2}(x) to rewrite cos^{4}(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 intergral 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
## ExercisesEvaluate the following integrals.1. cos ^{3}(x) sin^{2}(x) dx
2. sin ^{3}(x) cos^{14}(x) dx
## Answers to Above Exercises1. - (1/5)sin^{5}(x) + (1/3)sin^{3}(x)
2. (1/17)cos ^{17}(x) - (1/15)cos^{15}(x)
## More references and Linksintegrals and their applications in calculus. |