Integral $\int \frac{\cos(x)+\sin(2x)}{\sin(x)}\text{ d}x$

Question

I would like to evaluate $$\int \dfrac{\cos(x)+\sin(2x)}{\sin(x)}\text{ d}x\text{.}$$

This is from Stewart's Calculus text, section 7.2., #19. Please note that I don't have a solutions manual and I am not interested in a complete solution.

I would merely like some guidance on a first step on approaching this one, since I've gotten 9 similar problems correct, and I hit a brick wall on this one. My first thought was perhaps splitting the fraction like so: $$\int\cot(x)\text{ d}x + \int\dfrac{\sin(2x)}{\sin(x)}\text{ d}x$$ but this does not look helpful.

Answer 1

Substituting using the double-angle identity $$\sin (2x) = 2 \sin x \cos x$$ will transform the integrand into an expression that involves only $\sin x$ and $\cos x$, which suggests a particular substitution.