Question

please solve the qstn ​

Answer 1

Step-by-step explanation:

We have,

$\int \frac{1}{ \tan(x) \tan( \frac{\pi}{3} - x ) \tan( \frac{\pi}{3} + x ) }dx$

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

$= \int \frac{ \cos(x) \cos(x) }{ \sqrt{3} \sin(x) \sin(x) }dx$

$= \frac{1}{ \sqrt{3} } \int \cot^{2} (x) dx$

$= \frac{1}{ \sqrt{3} } \int (\cosec ^{2} (x) - 1)dx$

$= \frac{1}{ \sqrt{3} } \int \cosec^{2} (x) - \frac{1}{ \sqrt{3} } \int \: dx$

$= \frac{1}{ \sqrt{3} } ( - \cot(x) ) - \frac{1}{ \sqrt{3} } x + c$

$= - \frac{1}{ \sqrt{3} } ( \cot(x) + x) + c$