Question

Integrate the function $\frac{1}{\sqrt{(x-1)(x-2)}}$

Answer 1

$(x-1)(x-2)=x^2-3x+2=x^2-3x+\frac{9}{4}-\frac{9}{4}+2$
$={(x-\frac{3}{2})}^2-\frac{1}{4}={(x-\frac{3}{2})}^2-{\left(\frac{1}{2}\right)}^2$
$\therefore \int \frac{1}{\sqrt{(x-1)(x-2)}}dx=\int \frac{1}{\sqrt{{(x-\frac{3}{2})}^2-{\left(\frac{1}{2}\right)}^2}}dx$
Let $x-\frac{3}{2}=t\Rightarrow dx=dt$
$\Rightarrow \int \frac{1}{\sqrt{{(x-\frac{3}{2})}^2-{\left(\frac{1}{2}\right)}^2}}dx=\int \frac{1}{\sqrt{t^2-{\left(\frac{1}{2}\right)}^2}}dt$
$=\log \left|t\sqrt{t^2-(\frac{1}{2}{)}^2}\right|+C$
$=\log |(x-\frac{3}{2})+\sqrt{x^2-3x+2}|+C$