<s>[INST] Evaluate: $ \int \frac{\sin x}{\sin x - \cos x} dx $ Consider
$$ \int \frac{\sin x}{\sin x - \cos x} dx $$
Well I tried taking integrand as $ \frac{\sin x - \cos x + \cos x}{\sin x - \cos x} $ so that it becomes,
$$ 1 + \frac{\cos x}{\sin x - \cos x} $$
But does not helps.
I want different techniques usable here.
 [/INST] Hint:
In general for 
$\displaystyle\dfrac{d\left(\dfrac{a\sin x+b\cos x}{c\sin x+d\cos x}\right)}{dx}$
write numerator$(a\sin x+b\cos x)$ as  $A(c\sin x+d\cos x)+B\cdot\dfrac{d(c\sin x+d\cos x)}{dx}$ 
Compare the coefficients of $\sin x,\cos x$ to find $A,B$
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