Answer
The solution is $$\int\frac{\sin\theta}{3-2\cos\theta}d\theta=\frac{1}{2}\ln|3-2\cos\theta|+c.$$
Work Step by Step
To solve the integral $$\int\frac{\sin\theta}{3-2\cos\theta}d\theta$$ we will use substitution $3-2\cos\theta=t\Rightarrow 2\sin\theta d\theta=dt\Rightarrow \sin\theta d\theta=\frac{dt}{2}.$ Putting this into the integral we have:
$$\int\frac{\sin\theta}{3-2\cos\theta}d\theta=\int\frac{1}{t}\frac{dt}{2}=\frac{1}{2}\ln |t| +c,$$
where $c$ is arbitrary constant. Now we have to express our solution (which is in terms of $t$) in terms of $\theta$:
$$\int\frac{\sin\theta}{3-2\cos\theta}d\theta=\frac{1}{2}\ln |t| +c=\frac{1}{2}\ln|3-2\cos\theta|+c.$$
