# Evaluate the following integral. Integral from 0 to pi/4 of tan^2 x dx. (Give an exact answer in...

## Question:

Evaluate the following integral.

{eq}\int_{0}^{\frac{\pi}{4}} \tan^2 x \, \mathrm{d}x {/eq}

(Give an exact answer in terms of {eq}\pi {/eq}.)

## Integration:

The process of finding a function, given its derivative, is called anti-differentiation (or integration). If F'(x) = f(x), we say F(x) is an anti-derivative of f(x).

Use identity {eq}1+\tan^2 x = \sec^2x {/eq}

{eq}\displaystyle \int_{0}^{\frac{\pi}{4}} \tan^2 x \, \mathrm{d}x\\ \displaystyle \int_{0}^{\frac{\pi}{4}} (\sec^2 x-1) \, \mathrm{d}x\\ \displaystyle \int_{0}^{\frac{\pi}{4}} \sec^2 xdx- \int_{0}^{\frac{\pi}{4}} dx\\ \displaystyle =(\tan x)_{0}^{\pi/4} -(x)_{0}^{\pi/4}\\ \displaystyle =1-0-\frac{\pi}{4}-0\\ = 1-\frac{\pi}{4} {/eq}