int frac{1}{sin (x-a) sin x} ,d x= | vedclass

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(B) Let $I = \int \frac{1}{\sin (x-a) \sin x} \,d x$. Multiply and divide by $\sin a$: $I = \frac{1}{\sin a} \int \frac{\sin a}{\sin (x-a) \sin x} \,d

Vedclass · Accepted Answer

$\operatorname{cosec} a(\log |\frac{\sin (x-a)}{\sin x}|)+c$, where $c$ is a constant of integration.

Vedclass · Answer

(B) Let $I = \int \frac{1}{\sin (x-a) \sin x} \,d x$. Multiply and divide by $\sin a$: $I = \frac{1}{\sin a} \int \frac{\sin a}{\sin (x-a) \sin x} \,d x$. Express $\sin a$ as $\sin (x - (x-a))$: $I = \frac{1}{\sin a} \int \frac{\sin (x - (x-a))}{\sin (x-a) \sin x} \,d x$. Using the identity $\sin (A-B) = \sin A \cos B - \cos A \sin B$: $I = \frac{1}{\sin a} \int \frac{\sin x \cos (x-a) - \cos x \sin (x-a)}{\sin (x-a) \sin x} \,d x$. Split the integral: $I = \frac{1}{\sin a} \left[ \int \frac{\sin x \cos (x-a)}{\sin (x-a) \sin x} \,d x - \int \frac{\cos x \sin (x-a)}{\sin (x-a) \sin x} \,d x \right]$. $I = \frac{1}{\sin a} \left[ \int \cot (x-a) \,d x - \int \cot x \,d x \right]$. Integrating: $I = \frac{1}{\sin a} [\log |\sin (x-a)| - \log |\sin x|] + c$. $I = \operatorname{cosec} a \log |\frac{\sin (x-a)}{\sin x}| + c$.

$\int \frac{1}{\sin (x-a) \sin x} \,d x=$

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