Integral of 1+2cotx(cotx+cosecx) with respect to x is :

I=∫1+2cosecx cotx+2cot2x=∫cosec2x+2cosec xcotx+cot2x dx=∫(cosecx+cotx)dx

Integral of 1+2cotx(cotx+cosecx) with respect to x is :

I=∫1+2cosecx cotx+2cot2x=∫cosec2x+2cosec xcotx+cot2x dx=∫(cosecx+cotx)dx

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Integral of $\sqrt{1 + 2 \cot x (\cot x + \cos e c x)}$ with respect to $x$ is :

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$2 lncos \frac{x}{2} + c$

$2 lnsin \frac{x}{2} + c$

$\frac{1}{2} l n \sin \frac{x}{2} + c$

$l n \sin x - \ln (\cos e c x - \cot x) + c$

answer is B.

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$\begin{array}{l} I = \int \sqrt{1 + 2 \cos e c x \cot x + 2 \cot^{2} x} \\ = \int \sqrt{\cos e c^{2} x + 2 \cos e c x \cot x + \cot^{2} x d x} \\ = \int (\cos e c x + \cot x) d x \end{array}$

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Integral of 1+2cotx(cotx+cosecx) with respect to x is :

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Integral of $\sqrt{1 + 2 \cot x (\cot x + \cos e c x)}$ with respect to $x$ is :