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.