Question

Evaluate: $\int \frac{sin2x}{sin\left(x-\frac{\pi }{3}\right)sin\left(x+\frac{\pi }{3}\right)}dx$

• 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)

Let $I=\int \frac{sin2x}{sin\left(x-\frac{\pi }{3}\right)sin\left(x+\frac{\pi }{3}\right)}dx$. then,

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

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

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

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