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  4. Evaluate: ∫(sin 2x/sin(x -(π)3) sin(x+(π)3) dx

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Q. Evaluate: $\int\frac{sin \,2x}{sin\left(x -\frac{\pi}{3}\right) sin\left(x+\frac{\pi}{3}\right)} dx $

Integrals

Solution:

Let $ I = \int\frac{sin\, 2x}{sin\left(x -\frac{\pi}{3}\right) sin\left(x+\frac{\pi}{3}\right)} dx $. then,

$I = \int\frac{sin\left\{\left(x-\frac{\pi}{3}\right) + \left(x+\frac{\pi }{3}\right)\right\}}{sin\left(x-\frac{\pi }{3}\right) sin \left(x+\frac{\pi }{3}\right)} dx$

$\Rightarrow I = \int \frac{\left\{sin\left(x-\frac{\pi }{3}\right) cos \left(x+\frac{\pi }{3}\right) + cos\left(x-\frac{\pi }{3}\right) sin\left(x+\frac{\pi }{3}\right)\right\}}{sin\left(x-\frac{\pi }{3}\right) sin\left(x+\frac{\pi }{3}\right)} dx $

$\Rightarrow I=\int\left\{cot\left(x+\frac{\pi }{3}\right)+ cot\left(x-\frac{\pi }{3}\right)\right\}dx$

$\Rightarrow I = log\left|sin\left(x+\frac{\pi \:}{3}\right)\:\right|+log\:\left|sin\left(x-\frac{\pi \:}{3}\right)\:\right|\:+\:C$