Question

Evaluate the definite integral ${\int }_0^{\frac{\pi }{4}}\tan xdx$

Answer 1

Let $I={\int }_0^{\frac{\pi }{4}}\tan xdx$

$\Rightarrow \int \tan xdx=-\log |\cos x|=F\left(x\right)$

By second fundamental theorem of calculus, we obtain

$I=F\left(\frac{\pi }{4}\right)-F\left(0\right)$

$=-\log |\cos \frac{\pi }{4}|+\log |\cos 0|$

$=-\log \left|\frac{1}{\sqrt{2}}\right|+\log |1|$

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