`int_(pi//6)^(pi//3)1/(sin2x)dx\ ` is equal to `(log)_e3` b. `(log)_"e"sqrt(3)` c. `1/2log(-1)` d. `log(-1)`

Explanation

`log _{e} sqrt{3} `

`int_{frac{pi}{6}}^{frac{pi}{3}} frac{1}{sin 2 x} d x `

`=int_{frac{pi}{6}}^{frac{pi}{3}} cosec 2 x d x `

`=frac{1}{2} int_{frac{pi}{6}}^{frac{pi}{3}} 2 cosec 2 x d x `

`=frac{-1}{2}[log (cosec 2 x+cot 2 x)]_{frac{pi}{6}}^{frac{pi}{3}} `

`=frac{-1}{2}[-2 log sqrt{3}] `

`=log sqrt{3}`