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. Examples and exercises with solutions are included.
Examples with Detailed solutions
In what follows, C is the constant of integration.
Example 1
Evaluate the integral
\[ \int \sin^3(x) \cos^2(x) dx \]
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:
\( \displaystyle \int \sin^3(x) \cos^2(x) dx = \int \sin^2(x) \cos^2(x) \sin(x) dx \)
We now use the identity \( \sin^2(x) = 1 - \cos^2(x) \) and rewrite the given integral as follows:
\( \displaystyle \int \sin^3(x) \cos^2(x) dx = \int (1 - \cos^2(x)) \cos^2(x) \sin(x) dx \)
We now let \( u = \cos(x) \), hence \( du/dx = -\sin(x) \) or \( -du = \sin(x)dx \) and substitute in the given integral to obtain
\( \displaystyle \int \sin^3(x) \cos^2(x) dx = -\int (1 - u^2) u^2 du \)
Expand and calculate the integral on the right
\( \displaystyle \int \sin^3(x) \cos^2(x) dx = \int (u^4 - u^2) du \)
\( = \dfrac{1}{5}u^5 - \dfrac{1}{3}u^3 + C \)
Substitute \( u \) by \( \cos(x) \) to obtain
\[ \displaystyle \int \sin^3(x) \cos^2(x) dx = \dfrac{1}{5}\cos^5(x) - \dfrac{1}{3}\cos^3(x) + C \]
Example 2
Evaluate the integral
\[ \int \sin^{12}(x) \cos^5(x) dx \]
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:
\( \displaystyle \int \sin^{12}(x) \cos^5(x) dx = \int \sin^{12}(x) \cos^4(x) \cos(x) dx \)
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:
\( \displaystyle \int \sin^{12}(x) \cos^5(x) dx = \int \sin^{12}(x) (1 - \sin^2(x))^2 \cos(x) dx \)
We now let \( u = \sin(x) \), hence \( du/dx = \cos(x) \) or \( du = \cos(x)dx \) and substitute in the given integral to obtain
\( \displaystyle \int \sin^{12}(x) \cos^5(x) dx = \int u^{12} (1 - u^2)^2 du \)
Expand and calculate the integral on the right
\( \displaystyle \int \sin^{12}(x) \cos^5(x) dx = \int (u^{16} - 2u^{14} + u^{12}) du \)
\( = \dfrac{1}{17}u^{17} - \dfrac{2}{15}u^{15} + \dfrac{1}{13}u^{13} + C \)
Substitute \( u \) by \( \sin(x) \) to obtain
\( \displaystyle \int \sin^{12}(x) \cos^5(x) dx = \dfrac{1}{17}\sin^{17}(x) - \dfrac{2}{15}\sin^{15}(x) + \dfrac{1}{13}\sin^{13}(x) + C \)
Exercises
Evaluate the following integrals.
1. \( \displaystyle \int \cos^3(x) \sin^2(x) dx \)
2. \( \displaystyle \int \sin^3(x) \cos^{14}(x) dx \)
