Question

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

Answer 1

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

$=\int (\frac{23}{41}-\frac{2(5\cos x-4\sin x)}{41(5\cos x+4\sin x)})dx$

$=\int \frac{23}{41}-\frac{2(5\cos x-4\sin x)}{41(5\cos x+4\sin x)}dx$

$=\frac{23}{41}\int 1dx-\frac{2}{41}\int \frac{5\cos x-4\sin x}{5\cos x+4\sin x}dx$

$substitute$ $u=5\cos x+4\sin x\to du=(5\cos x-4\sin x)dx$

$=\frac{23}{41}x-\frac{2}{41}\int \frac{1}{u}du$

$=\frac{23}{41}x-\frac{2}{41}\ln u$

$=\frac{23}{41}x-\frac{2}{41}\ln (|5\sin x+4\cos x|)+C$