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
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