# Integrals Involving sin(x) with Even Power

Tutorial to find integrals involving even powers of sin(x), using reducing power formulas, are presented. Exercises with answers are at the bottom of the page. Integrating odd powers of sine is much simpler.

## Review Power Reducing Formulas
For n a positive integer, we have the following formulas in trigonometry that may be used to reduce the power of sin(x).
## Examples with Detailed SolutionsIn what follows, C is the constant of integration.
## Example 1Evaluate the integral^{2}(x) dx
Solution to Example 1:The main idea is to use the identity sin ^{2}x = (1/2)( 1 - cos(2x) ) to reduce the power and hence make the integral easily evaluated as follows:sin ^{2}(x) dx = (1/2)( 1 - cos(2x) ) dx
= (1/2)dx - (1/2) cos(2x) dx = (1/2) x - (1/2)*(1/2) sin(2x) = x / 2 - (1/4) sin(2x) + C
## Example 2Evaluate the integral^{2}(x) - 16 sin^{6}(x) ]dx
Solution to Example 2:Use the power reducing formulas to rewrite the integral as follows [ sin ^{2}(x) - 16 sin^{6}(x) ] dx
= [ (1/2)( 1 - cos(2x) ) - 16 (1/32)( 10 - 15cos(2x) + 6 cos(4x) - cos(6x) ) ]dx = (1/2) [ 1 - cos(2x) - 10 + 15cos(2x) - 6 cos(4x) + cos(6x) ] dx = (1/2) [ - 9 + 14cos(2x) - 6 cos(4x) + cos(6x) ] dx = (1/2) [ - 9 x + 7 sin(2x) - (3/2) sin(4x) + (1/6) sin(6x) ] + C
## ExercisesEvaluate the following integrals.(a) [ 8 sin ^{6}(x) - 2 sin^{2}(x) ]dx
(b) [ 4 sin ^{4}(x) + sin^{2}(x) ]dx## Answers to Above Exercises(a) (3/2) x - (11/8) sin(2x) + (3/8) sin(4x) - (1/24) sin(6x) + C(b) 2 x - (5/4)sin(2x) + (1/8) sin(4x) + C ## More References and linksintegrals and their applications in calculus. |