Integrals Involving sin(x) with Odd Power
Tutorial to find integrals involving odd powers of sin(x). Exercises with answers are at the bottom of the page.
Examples with Detailed Solutions
In what follows, C is the constant of integration.
Example 1
Evaluate the integral
sin3(x) dx
The main idea is to rewrite the power of sin(x) 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) dx = sin 2 (x) sin(x) dx
=
(1 - 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) dx = - (1 - u 2 ) du
sin 3 (x) dx = (1/3) u 3 - u + C
Substitute u by cos(x) to obtain
sin 3 (x) dx = (1/3)cos 3 (x) - cos(x) + C
Example 2
Evaluate the integralsin5(x) dx
Solution to Example 2:
Rewrite sin5(x) as follows sin5(x) = sin4(x) sin(x). Hence the given integral may be written as follows:
sin 5 (x) dx
= sin 4 (x) sin(x) dx
We now use the identity sin2(x) = 1 - cos2(x) to rewrite sin4(x) in terms of power of cos(x) and rewrite the given integral as follows:
sin 5 (x) dx
= (1 - cos 2 (x)) 2 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 5 (x) dx
= - (1 - u 2 ) 2 du
Expand and calculate the integral on the right
sin 5 (x) dx
= - (u 4 - 2u 2 + 1) du
= -(1/5)u 5 + (2/3)u 3 - u + C
and finally
sin 5 (x) dx
= -(1/5)cos 5 (x) + (2/3)cos 3 (x) - cos(x) + C
Exercises
Evaluate the following integrals.1.
sin 7 (x)dx
2.
sin 9 (x)dx
Answers to Above Exercises
1. (1/7)cos 7 (x) - (3/5)cos 5 (x) + cos 3 (x) - cos(x) + C2. -(1/9)cos 9 (x) + (4/7)cos 7 (x) - (6/5)cos 5 (x) + (4/3)cos 3 (x) - cos(x) + C
