# By using the properties of definite integrals, evaluate the integral $\int\limits_0^\frac{\Large \pi}{\Large 2}(2 \log \sin x-\log \sin 2x)\;dx$

This question has appeared in model paper 2012.

Toolbox:
• $m\log n=\log n^m$
• $\log m-\log n=\log\large\frac{m}{n}$
• $\sin 2x=2\sin x\cos x$
Step 1:
$I=\int_0^{\Large\frac{\pi}{2}}(2\log \sin x-\log \sin 2x)dx$
$\quad=\int_0^{\Large\frac{\pi}{2}}\log[\large\frac{\sin^2x}{\sin 2x}]$$dx \log m-\log n=\log\large\frac{m}{n} \quad=\int_0^{\Large\frac{\pi}{2}}\log \bigg(\large\frac{\sin^2x}{2\sin x\cos x}\bigg) \quad=\int_0^{\Large\frac{\pi}{2}}\log\bigg(\large\frac{\tan x}{2}\bigg)dx \quad=\int_0^{\Large\frac{\pi}{2}}\log(\tan x)-\log 2 dx \quad=\int_0^{\Large\frac{\pi}{2}}\log(\tan x)-\int_0^{\Large\frac{\pi}{2}}\log 2 dx Step 2: I=I_1-(\log 2[x]_0^{\Large\frac{\pi}{2}}------(1) \;\;=I_1-(\large\frac{\pi}{2}-0)$$\log 2$
Where $I_1=\int_0^{\Large\frac{\pi}{2}}\log(\tan x)dx$
$\int_0^{\Large\frac{\pi}{2}}\log(\tan(\large\frac{\pi}{2}-x))dx$
We know that $\int_0^af(x)dx=\int_0^af(a-x)dx$
$I_1=\int_0^{\large\frac{\pi}{2}}\log(\cot x)dx$
Step 3:
On adding the above equations we get,
$2I=\int_0^{\Large\frac{\pi}{2}}\log(\tan x)+\log(\cot x)]dx$
$\int_0^{\large\frac{\pi}{2}}\log(\tan x\cot x)dx$
$\log m+\log n=\log (mn)$
$I_1=0$
Step 4:
On substituting the value of $I_1$ in equ(1) we get
$I=I_1-(\large\frac{\pi}{2}$$-0)\log 2 I=0-(\large\frac{\pi}{2}$$-0)\log 2$
$\;\;=-\large\frac{-\pi}{2}$$\log 2$