Q56 / 100 · BITSAT 2024 (Memory Based Paper 3) · Mathematics
The value of definite integral $\int_0^{\pi / 2} \log (\tan x) d x$ is
- Correct answerA0
- B$\frac{\pi}{4}$
- C$\frac{\pi}{2}$
- D$\pi$
Solution
Let $I=\int_0^{\frac{\pi}{2}} \log (\tan x) d x$....(i)
By applying property,
$\begin{aligned}
\int_a^b f(x) d x & =\int_a^b f(a+b-x) d x \\
I & =\int_0^{\frac{\pi}{2}} \log \left(\tan \left(\frac{\pi}{2}-x\right)\right) d x \\
I & =\int_0^{\frac{\pi}{2}} \log (\cot x) d x\ldots(ii)
\end{aligned}$
Adding Eqs. (i) and (ii),
$\begin{aligned}
& 2 I=\int_0^{\frac{\pi}{2}}[\log (\tan x)+\log (\cot x)] d x \\
& 2 I=\int_0^{\frac{\pi}{2}} \log 1 d x=0
\end{aligned}$
Hence, the correct answer is option: 'A'.