Question

Evaluate the definite integrals.
${\int }_0^{\frac{\pi }{4}}tanxdx.$

Answer 1

${\int }_0^{\frac{\pi }{4}}tanxdx=[-log|cosx|{]}_0^{\frac{\pi }{4}}=-log\left|cos\frac{\pi }{4}\right|-(-log|cos0|)$
$=-log\left(\frac{1}{\sqrt{2}}\right)+log1=-log{2}^{-\frac{1}{2}}+0[\because \text{log 1 =0 and log m}{}^n\text{=n log m]}$=$\frac{1}{2}log2$