Answer
$\frac{1}{2}\ln|2\sin x+3|+C$
Work Step by Step
We have
$$\int \frac{\cos x}{2\sin x+3}dx =\frac{1}{2} \int \frac{2\cos x}{2\sin x+3}dx\\
=\frac{1}{2}\ln|2\sin x+3|+c.$$
where we used the facts that $(\sin u)'=\cos u $ and $\int \frac{u'}{u}=\ln u $.
