QJEEM24Maeea7: JEE Main 2024 Mathematics - Class 12 JEEM24Maeea7 Solution: Let . If , then is equal to

$\begin{aligned}& I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x \\& I(x)=\int \frac{6}{(\sin x-\cos x)^2} d x \\& =\int \frac{6 \sec ^2 x}{(\tan x-1)^2} d x\end{aligned}$

$\begin{aligned}& \text { Let } \tan x=t \Rightarrow \sec ^2 x d x=d t \\& =\int \frac{6 d t}{(t-1)^2} \\& =-\frac{6}{(t-1)}+c \\& =\frac{-6}{(\tan x-1)}+c \\& I(x)=\frac{6}{1-\tan x}+c \\& I(0)=3 \\& \frac{6}{1-\tan 0}+c=3 \\& c=-3\end{aligned}$

$\begin{aligned}I(x) & =\frac{6}{1-\tan x}-3 \\I\left(\frac{\pi}{12}\right) & =\frac{6}{1-\tan \left(\frac{\pi}{12}\right)}-3 \\& =\frac{6}{1-(2-\sqrt{3})}-3 \\& =\frac{6}{\sqrt{3}-1}-3 \\& =\frac{6-3 \sqrt{3}+3}{\sqrt{3}-1} \\& =\frac{9-3 \sqrt{3}}{\sqrt{3}-1} \\& =\frac{3 \sqrt{3}(\sqrt{3}-1)}{\sqrt{3}-1} \\& =3 \sqrt{3}\end{aligned}$