# Problem of the Week

## Updated at Dec 9, 2013 8:50 AM

This week's problem comes from the calculus category.

How can we solve for the integral of $$\frac{\tan{x}}{\sec^{4}x}$$?

Let's begin!

$\int \frac{\tan{x}}{\sec^{4}x} \, dx$

 1 Simplify the trigonometric functions.$\int \sin{x}\cos^{3}x \, dx$2 Use Integration by Substitution.Let $$u=\cos{x}$$, $$du=-\sin{x} \, dx$$3 Using $$u$$ and $$du$$ above, rewrite $$\int \sin{x}\cos^{3}x \, dx$$.$\int -{u}^{3} \, du$4 Use Power Rule: $$\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C$$.$-\frac{{u}^{4}}{4}$5 Substitute $$u=\cos{x}$$ back into the original integral.$-\frac{\cos^{4}x}{4}$6 Add constant.$-\frac{\cos^{4}x}{4}+C$Done-cos(x)^4/4+C