Question

The value of the integral $\int\limits_{0}^{\frac{\pi}{2}} log \left(tan\,x\right)dx=$

• A. 0
• B. 1
• C. $\frac{\pi}{2}$
• D. $\frac{\pi}{4}$

Answer 1

The Correct Option is A

Let I = $\int\limits_{0}^{\frac{\pi}{2}}$ log (tan x)dx ...(1)
Then, I = $\int\limits_{0}^{\frac{\pi}{2}} \, log\bigg[ tan \bigg(\frac{\pi}{2} - x\bigg)\bigg]dx$
$ \bigg[\because \, \, \int \limits_{0}^{a} \, f(x)dx \, = \, \int\limits_{0}^{a} \, f(a-x)dx\bigg]$
$\Rightarrow I = \int\limits_{0}^{\frac{\pi}{2}} \, log(cot x)dx$
$\Rightarrow \, \, I = \int\limits_{0}^{\frac{\pi}{2}} \, log\bigg(\frac{1}{tan \, x}\bigg)dx$
$\Rightarrow I = \int\limits_{0}^{\frac{\pi}{2}} \, log(tan x)^{-1} \, \, dx = -\int\limits_{0}^{\frac{\pi}{2}} \, \, log(tan x)dx$
$\Rightarrow$ I = -I $\Rightarrow $ 2I = 0 $\Rightarrow \, \, I \, = \, 0$
[Using e (1)]