Question

Evaluate :

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

Answer 1

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

$I=\int \frac{(3\sin x-2)\cos x}{5-(1-si{n}^{2}x)-4\sin x}dx$

Put $\sin x=t$

$\cos xdx=dt$

$I=\int \frac{3t-2}{5-(1-t^2)-4t}dt$

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

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

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

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

$I=3\log (t-2)-\frac{4}{t-2}+C$

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