Question

If $\int \frac{5\tan x}{\tan x-2}dx=x+a\ln |\sin x-2\cos x|+C$ for arbitrary constant of integration $C,$ then the value of $a$ is

Answer 1

$\int \frac{5\tan x}{\tan x-2}dx=\int \frac{5\sin x}{\sin x-2\cos x}dx=\int \frac{(\sin x-2\cos x)+2(\cos x+2\sin x)}{\sin x-2\cos x}dx=\int dx+2\int \frac{\cos x+2\sin x}{\sin x-2\cos x}dx=x+2\int \frac{\cos x+2\sin x}{\sin x-2\cos x}dx=x+2\ln |\sin x-2\cos x|+C$
On comparing, we get
$a=2$