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
\[ \int \sin^3(x) \, dx \]
Solution to Example 1:
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: \( \sin^3(x) = \sin^2(x) \sin(x) \). Hence the given integral may be written as follows:
\( \displaystyle \int \sin^3(x) \, dx = \int \sin^2(x) \sin(x) \, dx \)
Use the identity : \( \sin^2 x = 1 - \cos^2(x) \) to write
\( \displaystyle \int \sin^3(x) \, dx = \int (1 - \cos^2(x)) \sin(x) \, dx \)
We now let \( u = \cos(x) \), hence \( \dfrac{du}{dx} = -\sin(x) \) or \( -du = \sin(x) \, dx \) and substitute in the given integral to obtain
\( \displaystyle \int \sin^3(x) \, dx = - \int (1 - u^2) \, du \)
\( \displaystyle \int \sin^3(x) \, dx = \dfrac{1}{3} u^3 - u + C \)
Substitute \( u \) by \( \cos(x) \) to obtain
\[ \int \sin^3(x) \, dx = \dfrac{1}{3}\cos^3(x) - \cos(x) + C \]
Example 2
Evaluate the integral
\[ \int \sin^5(x) \, dx \]
Solution to Example 2:
Rewrite \( \sin^5(x) \) as follows \( \sin^5(x) = \sin^4(x) \sin(x) \). Hence the given integral may be written as follows:
\( \displaystyle \int \sin^5(x) \, dx = \int \sin^4(x) \sin(x) \, dx \)
We now use the identity \( \sin^2(x) = 1 - \cos^2(x) \) to rewrite \( \sin^4(x) \) in terms of power of \( \cos(x) \) and rewrite the given integral as follows:
\( \displaystyle \int \sin^5(x) \, dx = \int (1 - \cos^2(x))^2 \sin(x) \, dx \)
We now let \( u = \cos(x) \), hence \( \dfrac{du}{dx} = -\sin(x) \) or \( du = -\sin(x) \, dx \) and substitute in the given integral to obtain
\( \displaystyle \int \sin^5(x) \, dx = - \int (1 - u^2)^2 \, du \)
Expand and calculate the integral on the right
\( \displaystyle \int \sin^5(x) \, dx = - \int (u^4 - 2u^2 + 1) \, du \)
\( = -(\dfrac{1}{5}u^5 - \dfrac{2}{3}u^3 + u) + C \)
and finally
\[ \int \sin^5(x) \, dx = -(\dfrac{1}{5}\cos^5(x) - \dfrac{2}{3}\cos^3(x) + \cos(x)) + C \]
Exercises
Evaluate the following integrals.
1. \( \displaystyle \int \sin^7(x) \, dx \)
2. \( \displaystyle \int \sin^9(x) \, dx \)
