Question

$\int \frac{dx}{5\sin x+2\cos x+2}$ is equal to
(where $C$ is integration constant)

Answer 1

$I=\int \frac{dx}{5\sin x+2\cos x+2}$
Put $\tan \frac{x}{2}=t$
$\Rightarrow {\sec }^2\frac{x}{2}dx=2dt$
Now,
$I=\int \frac{\frac{2dt}{1+t^2}}{5\cdot \frac{2t}{1+t^2}+2\cdot \frac{1-t^2}{1+t^2}+2}[\because \sin x=\frac{2\tan \frac{x}{2}}{1+{\tan }^2\frac{x}{2}},\cos x=\frac{1-{\tan }^2\frac{x}{2}}{1+{\tan }^2\frac{x}{2}}]\Rightarrow I=\int \frac{2dt}{10t+2-2t^2+2+2t^2}\Rightarrow I=\int \frac{dt}{5t+2}\Rightarrow I=\frac{1}{5}\ln |5t+2|+C\therefore I=\frac{1}{5}\ln |5\tan \frac{x}{2}+2|+C$