Question

Evaluate each of the following integrals:

${\int }_0^{\frac{\mathrm{\pi }}{4}}\tan xdx$

Answer 1

$$\begin{gathered} {\int }_0^{\frac{\mathrm{\pi }}{4}}\tan xdx \\ ={\overline{)\mathrm{logsec}x}}_0^{\frac{\mathrm{\pi }}{4}} \\ =\mathrm{logsec}\frac{\mathrm{\pi }}{4}-\mathrm{logsec}0 \\ =\log \sqrt{2}-\log 1 \\ =\log 2^{\frac{1}{2}}-0 \\ =\frac{1}{2}\log 2 \end{gathered}$$