Let
\(I = \int \frac{2\sin2x - \cos x}{6cos^2x - 4\sin x}dx\)
\(= \int \frac{\cos x (4\sin x - 1)}{6 - 6\sin^2 x - 4\sin x }dx\)
Let \(\sin x =t\)
⇒ \(\cos x\,dx = dt\)
\(\therefore I = \int \frac{4t -1}{6-6t^2 -4t} dt\)
\(= -\frac13 \int \frac{12t + 4 - 7}{6t^2 + 4t -6}dt\)
\(= - \frac 13 \log (6t^2 + 4t - 6)+ \frac 7{18} \int \frac{dt}{t^2 + \frac 23 t- 1}+ C\)
\(= - \frac 13 \log (6t^2 + 4t - 6)+ \frac 7{18} \int \frac{dt}{(t + \frac 13)^2- \frac{10}9}+ C\)
\(= - \frac 13 \log (6t^2 + 4t - 6)+ \frac 7{18} \times \frac 1{2\frac{\sqrt {10}}3} \log \left|\frac{t + \frac 13 - \frac{\sqrt{10}}3}{t + \frac 13 + \frac{\sqrt{10}}3}\right|+C\)
\(= - \frac 13 \log (6t^2 + 4t - 6)+ \frac 7{12\sqrt{10}} \log \left|\frac{3t +1 - \sqrt{10}}{3t +1+\sqrt{10}}\right|+C\)
\(= - \frac 13 \log (6\sin^2 x + 4\sin x -6) + \frac 7{12\sqrt{10}}\log\left|\frac{3\sin x+ 1- \sqrt{10}}{3\sin x+ 1+ \sqrt{10}}\right|+C\)
