Evaluating $\int_{0}^{1}{\frac{3x^3 - x^2 + 2x - 4}{\sqrt{x^2 -3x+2}}dx}$ I've got one integration question which I first felt was not a hard nut to crack. But, as I proceeded, difficulties arose. This is the one:

$\displaystyle\int_{0}^{1}{\frac{3x^3 - x^2 + 2x - 4}{\sqrt{x^2 -3x+2}}dx}$

I went ahead simplifying the two expressions and ultimately I reached this step:
$\displaystyle\int_{0}^{1}{\sqrt{\frac{x-1}{x-2}} \ (3x^2 + 2x + 4) \ dx}$
I don't now what to do now, had I followed the correct pathway? Is there any other simpler method? 

You can try a hyperbolic substitution:
$$\sqrt{x^2-3x+2}=\sqrt{\Bigl(x-\frac{3\strut}2\Bigr)^{2}-\frac94+2}=\frac12\sqrt{(2x-3)^2-1},$$
so you can set, for $t\ge 0$,
$$2x-3=\cosh t\iff x=\frac{\cosh t+3}2,\qquad\mathrm dx=\frac12\sinh t\,\mathrm dt.$$
and the denominator of the integrand becomes $\;\frac12\sinh t$.
After you've simplified, you should obtain the integral of a cubic polynomial in $\cosh t$.
