If $\int\frac{3\cos x+2\sin x}{4\sin x+5\cos x}dx=Ax+B\log|4\sin x+5\cos x|+C$, then:

$A=\frac{23}{41},B=\frac{2}{41}$

$3\cos x + 2\sin x = l(4 \sin x + 5\cos x ) +m(4 \cos x - 5\sin x )$

$⇒l=\frac{23}{41}$ and $m=\frac{2}{41}$ ⇒ The given integral is

$\frac{23}{41}\int dx+\frac{2}{41}\int\frac{4\cos x-5\sin x}{4\sin x+5\cos x}dx=\frac{23x}{41}+\frac{2}{41}\log|4\sin x+5\cos x|+C$