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Question: Answered & Verified by Expert
$\int \sqrt{1+2 \cot x(\cot x+\operatorname{cosec} x)} d x=$
MathematicsIndefinite IntegrationTS EAMCETTS EAMCET 2019 (04 May Shift 2)
Options:
  • A $2 \log \left|\sin \frac{x}{2}\right|+c$
  • B $2 \log \left|\cos \frac{x}{2}\right|+c$
  • C $\log \left|\sin \frac{x}{2}+\cos \frac{x}{2}\right|+c$
  • D $2 \log |\sin x+\cos x|+c$
Solution:
2461 Upvotes Verified Answer
The correct answer is: $2 \log \left|\sin \frac{x}{2}\right|+c$
We have,
$$
\begin{aligned}
I & =\int \sqrt{1+2 \cot x(\cot x+\operatorname{cosec} x)} d x \\
I & =\int \sqrt{1+\cot ^2 x+2 \operatorname{cosec} x \cot x+\cot ^2 x} d x \\
I & =\int \sqrt{(\operatorname{cosec} x+\cot x)^2} d x \\
I & =\int(\operatorname{cosec} x+\cot x) d x \\
I & =\log (\operatorname{cosec} x-\cot x)+\log \sin x+c \\
I & =\log (\operatorname{cosec} x-\cot x) \sin x+c \\
I & =\log (1-\cos x)+c \\
\Rightarrow I & =\log \left(2 \sin ^2 \frac{x}{2}\right)+c \Rightarrow I=2 \log \left|\sin \frac{x}{2}\right|+c
\end{aligned}
$$

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