$\int \frac{3\cos x-2\sin x}{4\sin x+5\cos x}dx=$

$$\begin{gathered} \int \frac{a\cos x+b\sin x}{c\cos x+d\sin x}dx \\ =\left(\frac{ac+bd}{c^2+d^2}\right)x+\left(\frac{ad-bc}{c^2+d^2}\right)\log \left|\text{denominator}\right|+e \end{gathered}$$
G.t a=3, b=-2, c=4, d=5
$\Rightarrow \frac{7}{41}x+\frac{22}{41}\log |5\cos x+4\sin x|+C$