Sine and Cosine Power Integrals
Simple overview of integrals that are of the form \(\int \sin^n(x)\cos^m(x)dx\).
Main Idea
Consider the possible integrals of the following form
\[\int \sin^n(x)\cos^m(x)dx\]where $n$ and $m$ are whole numbers greater than or equal to $1$.
There are four cases you can follow to calculate the integral that follow.
Watch this section: four cases at 0:12.
Case 1: Odd Sine Power
Consider the following
\[\int \sin^5(x)\cos^2(x)dx\]where sine power is odd.
First put one copy of $\sin(x)$ to the side
\[\sin^5(x)=\sin^4(x)\sin(x)\]Then the remaining even power of sine can be rewritten using
\[\sin^2(x)=1-\cos^2(x).\]That gives
\[\sin^4(x)=\left(\sin^2(x)\right)^2 =\left(1-\cos^2(x)\right)^2\]So the integral becomes
\[\int \left(1-\cos^2(x)\right)^2\cos^2(x)\sin(x)dx\]Now we have $sin(x)$ off to the side, which will help us do u-substitution.
Let
\[u=\cos(x)\]so
\[du=-\sin(x)dx\]and
\[\sin(x)dx=-du\]After substituting, the video gets
\[-\int (1-u^2)^2u^2 du\]Expand the integral to use the power rule
\[(1-u^2)^2u^2=u^2-2u^4+u^6\]Then integrate term by term
\[-\int \left(u^2-2u^4+u^6\right)\,du =-\left(\frac{u^3}{3}-\frac{2u^5}{5}+\frac{u^7}{7}\right)+C\]Substitute $u=\cos(x)$ back in
\[-\frac{\cos^3(x)}{3} +\frac{2\cos^5(x)}{5} -\frac{\cos^7(x)}{7}+C\]Watch this section: sine odd, cosine even at 0:28.
Case 2: Odd Cosine Power
Now, what happens when cosine is odd?
\[\int \sin^4(x)\cos^3(x) dx\]Since cosine has the odd power, we put one $\cos(x)$ to the side
\[\cos^3(x)=\cos^2(x)\cos(x)\]Then it rewrites the remaining cosine power with
\[\cos^2(x)=1-\sin^2(x)\]The integral becomes
\[\int \sin^4(x)\left(1-\sin^2(x)\right)\cos(x) dx\]With one $cos(x)$ to the side we are set up for u-substitution.
Let
\[u=\sin(x)\]so
\[du=\cos(x) dx\]Substitute and simplify
\[\int u^4(1-u^2) du =\int (u^4-u^6) du\]Integrating gives
\[\frac{u^5}{5}-\frac{u^7}{7}+C\]Return to $x$
\[\frac{\sin^5(x)}{5}-\frac{\sin^7(x)}{7}+C\]Watch this section: sine even, cosine odd at 3:26.
Case 3: Both Powers Odd
What do we do when both powers are odd?
\[\int \sin^3(x)\cos^3(x) dx\]If both powers are odd, you can do either case 1 or case 2 to find the integral.
Let’s keep one $\cos(x)$ on the side for this one.
We can rewrite
\[\cos^3(x)=\cos^2(x)\cos(x)\]Then it converts the remaining cosine square:
\[\cos^2(x)=1-\sin^2(x)\]So the integral becomes
\[\int \sin^3(x)\left(1-\sin^2(x)\right)\cos(x) dx\]Let
\[u=\sin(x), \qquad du=\cos(x) dx\]Then
\[\int u^3(1-u^2) du =\int (u^3-u^5) du\]Integrate
\[\frac{u^4}{4}-\frac{u^6}{6}+C\]Substitute back
\[\frac{\sin^4(x)}{4}-\frac{\sin^6(x)}{6}+C\]Watch this section: both powers odd at 5:17.
Case 4: Both Powers Even
What happens when both powers are even?
\[\int \sin^2(x)\cos^2(x) dx\]When both powers are even, there is no single $\sin(x)dx$ or $\cos(x)dx$ piece to save for an immediate u-substitution.
We switch to a power reduction method
\[\sin^2(x)=\frac{1-\cos(2x)}{2}\]and
\[\cos^2(x)=\frac{1+\cos(2x)}{2}\]Substitute both
\[\int \sin^2(x)\cos^2(x) dx =\frac{1}{4}\int \left(1-\cos(2x)\right)\left(1+\cos(2x)\right) dx\]Use the difference of squares
\[\left(1-\cos(2x)\right)\left(1+\cos(2x)\right) =1-\cos^2(2x)\]So
\[\frac{1}{4}\int \left(1-\cos^2(2x)\right) dx\]There is still an even power, so reduce again
\[\cos^2(2x)=\frac{1+\cos(4x)}{2}\]After simplifying, we get
\[\frac{1}{8}\int \left(1-\cos(4x)\right) dx\]Now integrate:
\[\frac{1}{8}\left(x-\frac{\sin(4x)}{4}\right)+C\]The final answer is
\[\frac{x}{8}-\frac{\sin(4x)}{32}+C\]Watch this section: both powers even at 6:44.
Video Summary
The video ends by returning to the same decision tree:
| Case | What to do |
|---|---|
| $n$ odd, $m$ even | Save $\sin(x)$, convert sine powers to cosine, use $u=\cos(x)$. |
| $n$ even, $m$ odd | Save $\cos(x)$, convert cosine powers to sine, use $u=\sin(x)$. |
| $n$ odd, $m$ odd | Pick either odd-power method. |
| $n$ even, $m$ even | Use power reduction identities. |
The quick habit is: check the exponents first. Once you know which powers are odd or even, the next move is much less mysterious.
Watch this section: summary at 10:29.
Timestamp Guide
Related Calculus Work
This lesson uses u-substitution in the first three cases. For a slower review of that method, see the u-substitution topic page.
The examples are indefinite integrals, so every final answer includes $+C$. For that background, review the constant of integration notes.
The video description also points students toward the JoeCMath integration playlist: watch the integration playlist on YouTube.