Question

Solve:
$\int \frac{4\sin x-38\cos x}{5\sin x-11\cos x}$

Answer 1

$\int \frac{4\sin x-38\cos x}{5\sin x-11\cos x}dx$

$4\sin x-38\cos x=a(5\sin x-11\cos x)+b(5\cos x+11\sin x)$

$4\sin x-38\cos x=(5a+11b)\sin x+(-11a+5b)\cos x$

$5a+11b=4$ ....... (i) $\times 11$

$-11a+5b=-38$ .....(ii) $\times 5$

$b=-1$

$5a+11\times (-1)=4$

$5a-114$

$-5a=-15$

$a=3$

$\int \frac{(4\sin x-38\cos x)}{4\sin x-11\cos x)}=\int \frac{3(5\sin x-11\cos x)}{(5\sin x-11\cos x)}-\frac{(5\cos x+11\sin x)}{(5\sin x-11\cos x)}$

$\int \frac{4\sin x-38\cos x)}{4\sin x-11\cos x)}=\int 3dx-\int \frac{5\cos x+11\cos x}{5\sin x-11\cos x}$

$=3x-\ln (5\sin x-11\cos x)+C$