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 tan⁡xsec⁡4x\frac{\tan{x}}{\sec^{4}x}?

Let's begin!



∫tan⁡xsec⁡4x dx\int \frac{\tan{x}}{\sec^{4}x} \, dx

1
Simplify the trigonometric functions.
∫sin⁡xcos⁡3x dx\int \sin{x}\cos^{3}x \, dx

2
Use Integration by Substitution.
Let u=cos⁡xu=\cos{x}, du=−sin⁡x dxdu=-\sin{x} \, dx

3
Using uu and dudu above, rewrite ∫sin⁡xcos⁡3x dx\int \sin{x}\cos^{3}x \, dx.
∫−u3 du\int -{u}^{3} \, du

4
Use Power Rule: ∫xn dx=xn+1n+1+C\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C.
−u44-\frac{{u}^{4}}{4}

5
Substitute u=cos⁡xu=\cos{x} back into the original integral.
−cos⁡4x4-\frac{\cos^{4}x}{4}

6
Add constant.
−cos⁡4x4+C-\frac{\cos^{4}x}{4}+C

Done
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