$\int \frac{3\sin x-5\cos x}{7\cos x+2\sin x}dx=$

$\int \frac{3\sin x-5\cos x}{7\cos x+2\sin x}dx$

$\int \frac{a \cos x+b\sin x}{c \cos x+d \sin x}dx=Ax+B\log \left| c\cos x+d \sin x\right|$

where $A=\frac{ac+bd}{c^2+d^2}; B=\frac{ad-bc}{c^2+d^2}$

$I=\int \frac{-5\cos x+3\sin x}{7\cos x+2\sin x}dx$

Here $a=-5, b=3, c=7, d=2$

$A=\frac{ad-bc}{c^2+d^2}=\frac{-10-21}{49+4}=\frac{-31}{53}$

$I=Ax+B \log \left|7 \cos x+2 \sin x\right|+C$

$=\frac{-29}{53}x-\frac{31}{53}\log \left|7\cos x+2\sin x\right|+C$