Question

If $\int \frac{3\cos x+2\sin x}{4\sin x+5\cos x}dx=ax+b\ln |4\sin x+5\cos x|+C$, then which of the following is/are correct?
(where $a,b$ are fixed constants and $C$ is integration constant)

Answer 1

Let
$3\cos x+2\sin x=A(4\sin x+5\cos x)+B\frac{d}{dx}(4\sin x+5\cos x)=A(4\sin x+5\cos x)+B(4\cos x-5\sin x)$
Comparing the coefficients of $\sin x$ and $\cos x$, we get
$4A-5B=25A+4B=3$
Solving, we get
$A=\frac{23}{41}$ and $B=\frac{2}{41}$
Thus the given integral reduces to
$I=\frac{23}{41}\int dx+\frac{2}{41}\int \frac{4\cos x-5\sin x}{4\sin x+5\cos x}dx=\frac{23}{41}x+\frac{2}{41}\ln |4\sin x+5\cos x|+C$
Thus,
$a-2b=\frac{19}{41}ab=\frac{46}{1681}$