Evaluate the definite integral in Exercise: _ 0 ^ 4 dx - Vidyadip

Question

Evaluate the definite integral in Exercise:
$\int\limits_{0}^{\frac{\pi}{4}}\tan\text{x}\ \text{dx}$

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Answer

$\text{Let}\ \text{I}=\int\limits_{0}^\frac{\pi}{4}\tan\text{x}\ \text{dx}$

$\int\tan\text{x}\ \text{dx}=-\text{log}|\cos\text{x|}=\text{F}\text{(x)}$

By second fundamental theoram of calculus, we obtain

$\text{I}=\text{F}\bigg(\frac{\pi}{4}\bigg)-\text{F}(0)$

$=-\text{log}|\cos\frac{\pi}{4}|+\text{log}|\cos0|$

$=-\text{log}\big|\frac{1}{\sqrt{2}}\big|+\text{log}|1|$

$=\text{log}(2)^\frac{1}{2}$

$=\frac{1}{2}\text{log}2$

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