Question

If $f\left(x\right)=cosec(x-\pi /3)cosec(x-\pi /6)$, then the value of ${\int }_0^{\pi /2}f\left(x\right)dx$ is

• A. $4\log \sqrt{3}$
• B. $-4\log \sqrt{3}$
• C. $2\log \sqrt{3}$
• D. $\frac{1}{4}$

Answer 1

Answer: (B)

$-4\log \sqrt{3}$

$\frac{1}{2}=\sin \frac{\pi }{6}=\sin [(\frac{\pi }{3}-x)+(x-\frac{\pi }{6})]$
$\Rightarrow 1=-2\sin (x-\frac{\pi }{3})\cos (x-\frac{\pi }{6})+2\sin (x-\frac{\pi }{6})\cos (x-\frac{\pi }{3})$
Multiply both sides by $\csc (x-\frac{\pi }{3})\csc (x-\frac{\pi }{6})$
And as $\int \cot xdx=\log \left|\sin x\right|$
Then ${\int }_0^{\frac{\pi }{2}}f\left(x\right)dx=-\log 3+2\log \frac{1}{\sqrt{3}}=-4\log \sqrt{3}$