int frac{d x}{sin left(x-frac{pi}{3}right) co | vedclass

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(A) Let $I = \int \frac{dx}{\sin(x - \frac{\pi}{3}) \cos x}$.Using the formula $\sin(A - B) = \sin A \cos B - \cos A \sin B$,we have $\sin(x - \frac{\

Vedclass · Accepted Answer

$2 \log \left| 	an x - \sqrt{3} ight| + C$

Vedclass · Answer

(A) Let $I = \int \frac{dx}{\sin(x - \frac{\pi}{3}) \cos x}$.Using the formula $\sin(A - B) = \sin A \cos B - \cos A \sin B$,we have $\sin(x - \frac{\pi}{3}) = \sin x \cos(\frac{\pi}{3}) - \cos x \sin(\frac{\pi}{3}) = \frac{1}{2} \sin x - \frac{\sqrt{3}}{2} \cos x$.Substituting this into the integral:$I = \int \frac{dx}{(\frac{1}{2} \sin x - \frac{\sqrt{3}}{2} \cos x) \cos x} = \int \frac{dx}{\cos^2 x (\frac{1}{2} 	an x - \frac{\sqrt{3}}{2})} = \int \frac{\sec^2 x dx}{\frac{1}{2} (	an x - \sqrt{3})} = 2 \int \frac{\sec^2 x dx}{	an x - \sqrt{3}}$.Let $u = 	an x - \sqrt{3}$,then $du = \sec^2 x dx$.$I = 2 \int \frac{du}{u} = 2 \log |u| + C = 2 \log |	an x - \sqrt{3}| + C$.

$\int \frac{d x}{\sin \left(x-\frac{\pi}{3}\right) \cos x} = $

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