int frac{operatorname{cosec} x , dx}{cos^2(1 | vedclass

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(A) Let $I = \int \frac{\operatorname{cosec} x \, dx}{\cos^2(1 + \log 	an \frac{x}{2})}$.Let $t = 1 + \log(	an \frac{x}{2})$.Differentiating both si

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$	an(1 + \log(	an \frac{x}{2})) + c$,where $c$ is the constant of integration

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(A) Let $I = \int \frac{\operatorname{cosec} x \, dx}{\cos^2(1 + \log 	an \frac{x}{2})}$.Let $t = 1 + \log(	an \frac{x}{2})$.Differentiating both sides with respect to $x$,we get:$dt = \frac{1}{	an \frac{x}{2}} \cdot \sec^2 \frac{x}{2} \cdot \frac{1}{2} \, dx$.Using the identity $	an \frac{x}{2} = \frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}$ and $\sec^2 \frac{x}{2} = \frac{1}{\cos^2 \frac{x}{2}}$,we have:$dt = \frac{\cos \frac{x}{2}}{\sin \frac{x}{2}} \cdot \frac{1}{\cos^2 \frac{x}{2}} \cdot \frac{1}{2} \, dx = \frac{1}{2 \sin \frac{x}{2} \cos \frac{x}{2}} \, dx$.Since $\sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2}$,we get $dt = \frac{1}{\sin x} \, dx = \operatorname{cosec} x \, dx$.Substituting these into the integral:$I = \int \frac{1}{\cos^2 t} \, dt = \int \sec^2 t \, dt$.Integrating,we get $I = 	an t + c$.Substituting back the value of $t$,we get $I = 	an(1 + \log(	an \frac{x}{2})) + c$.

$\int \frac{\operatorname{cosec} x \, dx}{\cos^2(1 + \log \tan \frac{x}{2})} = $

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