Q102 / 180 · COMEDK 2018 · Mathematics

$\int_{0}^{\pi / 2} \log (\tan x) d x=$

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

Solution

Let $\quad I=\int_{0}^{\pi / 2} \log (\tan x) d x$ $\Rightarrow \quad I=\int_{0}^{\pi / 2} \log \tan \left(\frac{\pi}{2}-x\right) d x$ $\Rightarrow \quad I=\int_{0}^{\pi / 2} \log \cot x d x$ $\Rightarrow \quad I=\int_{0}^{\pi / 2} \log \frac{1}{\tan x} d x$ $\Rightarrow \quad I=-\int_{0}^{\pi / 2} \log \tan x d x$ $\Rightarrow \quad I=-I \Rightarrow 2 I=0 \Rightarrow I=0$

Hence, the correct answer is option: 'B'.

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