Question

Evaluate the integral
${\int }_0^{\pi /2}\log \left(\tan x\right)dx$

• A. $0$
• B. $\pi /2$
• C. $\pi /3$
• D. $\pi /4$

Answer 1

The correct option is A $0$

$I={\int }_0^{\frac{\pi }{2}}log\left(tanx\right)dx$

${\int }_0^{a}f(x{)}^{dx}={\int }_0^{a}f(a-x)dx$

$I={\int }_0^{\frac{\pi }{2}}log\left(cotx\right)dx$

${\int }_0^{\frac{\pi }{2}}log\left(\frac{1}{tanx}\right)dx=-{\int }_0^{\frac{\pi }{2}}log\left(tanx\right)dx=I$

$2I=0$

$I=0$