Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
$\int_0^{\pi / 2} \log (\tan x) d x=$
A
$\frac{\pi}{2} \log _e 2$
B
$-\frac{\pi}{2} \log _{ e } 2$
C
$\pi \log _e 2$
✓
$0$
✓
Answer
Correct option: D.
$0$
(D) $\int_0^{\frac{\pi}{2}} \log \tan x d x=\int_0^{\frac{\pi}{2}} \log \left(\frac{\sin x}{\cos x}\right) d x$ $\begin{array}{l}=\int_0^{\frac{\pi}{2}} \log \sin x d x-\int_0^{\frac{\pi}{2}} \log \cos x d x \\ =\int_0^{\frac{\pi}{2}} \log \sin x d x-\int_0^{\frac{\pi}{2}} \log \sin x d x\end{array}$ $\ldots\left[\because \int_0^{ a } f (x) d x=\int_0^{ a } f ( a -x) d x\right]$ $=0$
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