How to take an integral using half angle trigonometric substitution. So i have this question which is asking to take the integral using a predefined trigonometric substitution which is $$u=\tan\frac{x}{2}$$
and the integral equation is $$\int\frac{\sin x\ dx}{(6\cos x-2)(3-2\sin x)}$$ How would i go on about this problem? Because to begin with i do not know how i would even use the given substitution method. Any help is appreciated thank you.

We have $$x=2arctan(u)$$, that gives $$dx=\frac{2}{u^2+1}du$$
Furthermore, we have $$cos(x)=\frac{1-u^2}{1+u^2}$$
and $$sin(x)=\frac{2u}{u^2+1}$$
Inserting those terms, you get a rational function of $u$. Try it from here.
