This line seems incorrect: $$\int_0^x{\frac{1}{(a-x)(b-x)}\,dx}$$ $$=\left[\frac{1}{b-a}\left(\ln{\frac{1}{a-x}}-\ln{\frac{1}{b-x}}\right)\right]_0^x$$ $$=\frac{1}{b-a}\left(\ln{\frac{a}{a-x}}-\ln{\frac{b}{b-x}}\right)$$