int frac{csc x}{log tan frac{x}{2}} , dx = | vedclass

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Question

(A) Let $I = \int \frac{\csc x}{\log 	an \frac{x}{2}} \, dx$. Substitute $t = \log 	an \frac{x}{2}$. Then,differentiating with respect to $x$,we get

Vedclass · Accepted Answer

$\log \left( \log 	an \frac{x}{2} ight) + c$

Vedclass · Answer

(A) Let $I = \int \frac{\csc x}{\log 	an \frac{x}{2}} \, dx$. Substitute $t = \log 	an \frac{x}{2}$. Then,differentiating with respect to $x$,we get: $\frac{dt}{dx} = \frac{1}{	an \frac{x}{2}} \cdot \sec^2 \frac{x}{2} \cdot \frac{1}{2} = \frac{\cos \frac{x}{2}}{\sin \frac{x}{2}} \cdot \frac{1}{\cos^2 \frac{x}{2}} \cdot \frac{1}{2} = \frac{1}{2 \sin \frac{x}{2} \cos \frac{x}{2}} = \frac{1}{\sin x} = \csc x$. Thus,$\csc x \, dx = dt$. Substituting these into the integral,we get: $I = \int \frac{1}{t} \, dt = \log |t| + c$. Replacing $t$ with $\log 	an \frac{x}{2}$,we obtain: $I = \log \left| \log 	an \frac{x}{2} ight| + c$.

$\int \frac{\csc x}{\log \tan \frac{x}{2}} \, dx = $

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