# 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 $\int \sin^3(x) \, dx$ 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. $$\displaystyle \int \sin^9(x) \, dx$$

1. $$\dfrac{1}{7}\cos^7(x) - \dfrac{3}{5}\cos^5(x) + \cos^3(x) - \cos(x) + C$$
2. $$-\dfrac{1}{9}\cos^9(x) + \dfrac{4}{7}\cos^7(x) - \dfrac{6}{5}\cos^5(x) + \dfrac{4}{3}\cos^3(x) - \cos(x) + C$$