# 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$Donesin(x)-sin(x)^3/3+C