Question

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

• A. $log\left|sin\left(x+\frac{\pi }{3}\right) \right|-log \left|sin\left(x-\frac{\pi }{3}\right) \right| + C$
• B. $log\left|sin\left(x+\frac{\pi }{3}\right) \right|+log \left|sin\left(x-\frac{\pi }{3}\right) \right| + C$
• C. $log\left|sin\left(x-\frac{\pi }{3}\right) \right|-log \left|sin\left(x+\frac{\pi }{3}\right) \right| + C$
• D. None of these

Answer 1

Answer: (B)

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

Answer 2

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