Question

$\int \frac{\left(3\sin x-2\right)\cos x}{5-{\cos }^{2}x-4\sin x}dx$

Answer 1

Given equation is,

$I=\int \frac{(3\sin x-2)\cos x}{5-{\cos }^{2}x-4\sin x}dx$

$=\int \frac{(3\sin x-6+4)\cos x}{4+1-{\cos }^{2}x-4\sin x}dx$

$=\int \frac{\left[3\cos x(\sin x-2)\right]+4\cos x}{4+{\sin }^{2}x-4\sin x}dx$

$=\int \frac{\left[3\cos x(\sin x-2)\right]+4\cos x}{{(\sin x-2)}^2}dx$

Let,

$t=\sin x$

$dt=\cos xdx$

$=\int \frac{[3(t-2)+4]}{{(t-2)}^2}dt$

$=\int \frac{3}{(t-2)}dt+\int \frac{4}{{(t-2)}^2}dt$

$=3\log |t-2|+4\frac{1}{-2+1}\times \frac{1}{t-2}+c$

$=3\log |2-t|-\frac{4}{t-2}+c$

$=3\log |2-\sin x|+\frac{4}{2-\sin x}+c$

Hence, $3\log |2-\sin x|+\frac{4}{2-\sin x}+c$