$\dfrac{1}{\sin \dfrac{\pi}{3}} \int \dfrac{\sin [(x+\dfrac{\pi}{3})-x]}{\sin x \sin (x+\dfrac{\pi}{3})}$
$=\dfrac{2}{\sqrt 3} \int \dfrac{\sin (x+\dfrac{\pi}{3}) \cos x -\cos (x+\dfrac{\pi}{3}) \sin x}{\sin x \sin (x+\dfrac{\pi}{3})}$
$=\dfrac{2}{\sqrt 3} \int \bigg ( \cot x -\cot (x+\dfrac{\pi}{3}) \bigg ] dx =\dfrac{2}{\sqrt 3} \bigg (\ln |sin (x+\dfrac{\pi}{3})| -\ln | \sin x| \bigg )+C$
$=\dfrac{2}{\sqrt 3} \ln \bigg | \dfrac{sin (x+\dfrac{\pi}{3})}{\sin x} \bigg | +C$ tự thay cận nhé
Trong đó có sử dụng $\int \cot (x+a) dx =\ln |\sin (x+a)|+C$ bạn tự làm nhé