Problem of the Week

Updated at Dec 23, 2013 12:50 PM

This week's problem comes from the calculus category.

How can we solve for the integral of \(\cos^{3}x\)?

Let's begin!



\[\int \cos^{3}x \, dx\]

1
Use Pythagorean Identities: \(\cos^{2}x=1-\sin^{2}x\).
\[\int (1-\sin^{2}x)\cos{x} \, dx\]

2
Use Integration by Substitution.
Let \(u=\sin{x}\), \(du=\cos{x} \, dx\)

3
Using \(u\) and \(du\) above, rewrite \(\int (1-\sin^{2}x)\cos{x} \, dx\).
\[\int 1-{u}^{2} \, du\]

4
Use Power Rule: \(\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C\).
\[u-\frac{{u}^{3}}{3}\]

5
Substitute \(u=\sin{x}\) back into the original integral.
\[\sin{x}-\frac{\sin^{3}x}{3}\]

6
Add constant.
\[\sin{x}-\frac{\sin^{3}x}{3}+C\]

Done
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