Added angle formula to solve this indefinite integral $\int\frac{2\cos x-\sin x}{3\sin x+5\cos x }\,dx$ Starting from this very nice question Integrate $\int\frac{2\cos x-\sin x}{3\sin x+5\cos x }\,dx$ and the relative answers, I would to understand because this integral $$\int\frac{2\cos x-\sin x}{3\sin x+5\cos x }\,dx \tag 1$$
must be split thus:
$$\int \frac{2\cos{x}-\sin{x}}{3\sin{x}+5\cos{x}} \; dx=\color{red}{\int A\left(\frac{3\sin{x}+5\cos{x}}{3\sin{x}+5\cos{x}}\right) +B \left(\frac{ 3\cos{x}-5\sin{x}}{3\sin{x}+5\cos{x}}\right)\; dx}$$
or it can be splitted in a different way.
Using the added angle formula (for numerator and denominator of the $(1)$) $$a\sin x+b\cos x=\lambda \sin (x+\phi)$$ if $\lambda=\sqrt{a^2+b^2}$ and $\tan \phi=b/a \ $ or $$a\sin x+b\cos x=\lambda \cos (x+\varphi)$$ with $\tan \varphi=-a/b \ $ is it possible to obtain the same result?

$I=\int\frac{2\cos x-\sin x}{3\sin x+5\cos x }\,dx $ 
$2\cos x-\sin x=\sqrt5 \cos(x+a) $, where $\tan a=1/2$ and $ 3\sin x+5\cos x=\sqrt{34}\cos(x-b) $, where $\tan b=3/5$ 
$I=\int \frac{\sqrt 5 \cos(x+a) } {\sqrt{34}\cos(x-b)} \, dx$ Substitute $t=x-b$ so that 
$I=\sqrt{5/34}\int \frac{\cos(t+a+b) } {\cos t}\, dt$
The integrand is now: $\cos(a+b) - \tan t\sin(a+b) $ Can you take it from here?
