How to integrate a rational function whose numerator and denominator are first order and second order polynomials?

Question

I know that

$$\int \frac{x-a}{x(1-x)}dx = -[a\ln x+ (1-a) \ln(1-x)]+const$$

but how can I prove it?

Answer 1

Let $$\frac{x-a}{x(1-x)}=\frac Ax + \frac B{1-x}$$

At $x=0$, $A = \frac{-a}{1} = -a$

At $x=1$, $B = \frac{1-a}{1} = 1-a$

So we have $$\frac{-a}{x} + \frac{1-a}{1-x}$$

Could you proceed further?