Integral of $\int \frac{\cos x+\sin 2x}{\sin x}$ I am trying to find the integral of $$\int \frac{\cos x+\sin 2x}{\sin x}$$
$$\int \frac{\cos x}{\sin x} + \int \frac{\sin 2x}{\sin x}$$
$$\int  \tan x + \int \frac{\sin 2x}{\sin x}$$
I think I am suppose to have the integral of tanx memorized so I will put that to the side for now.
$$\int \frac{\sin 2x}{\sin x}$$
I do not know what to do with this since I can't make a u subsitution or anything else so I will just randomly use the double angle identity I have memorized.
$$\int \frac{2\sin x\cos x}{\sin x}$$
$$\int 2\cos x$$
$$2 \int \cos x$$
$$2\sin x + \int  \tan x$$
$$2\sin x + \ln|\sec x| + c$$
This is of course wrong.

Hint 
Recall the chain rule $$g(f(x))'=g'(f(x))\cdot f'(x)$$
and note that
$$\frac{\cos x }{\sin x} =\frac{f'(x) }{f(x)} =\frac{f'(x) }{f(x)}=\frac{1 }{f(x)}\cdot f'(x)$$
