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.
In what follows, C is the constant of integration.
Example 1: Evaluate the integral
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 2: Evaluate the integral
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
Exercises: Evaluate the following integrals.
