Question

Integrate the following functions.
$\int \frac{1}{\sqrt{(x-1)(x-2)}}dx.$

Answer 1

Let $I=\int \frac{1}{\sqrt{(x-1)(x-2)}}dx=\int \frac{1}{\sqrt{x^2-3x+2}}dx$
$=\int \frac{1}{\sqrt{x^2-3x+(\frac{3}{2}{)}^2-(\frac{3}{2}{)}^2+2}}dx=\int \frac{1}{\sqrt{(x+\frac{3}{2}{)}^2+\frac{-9+8}{4}}}=\int \frac{1}{\sqrt{(x-\frac{3}{2}{)}^2-(\frac{1}{2}{)}^2}}dx$
Let $x-\frac{3}{2}=t\Rightarrow dx=dt$
$\therefore I=\int \frac{1}{\sqrt{t^2-(\frac{1}{2}{)}^2}}dt=log|t+\sqrt{t^2-\frac{1}{4}}|+C[\because \int \frac{dx}{\sqrt{x^2-a^2}}=log|x+\sqrt{x^2-a^2}|]=log|x-\frac{3}{2}+\sqrt{(x-\frac{3}{2}{)}^2}-\frac{1}{4}|+C=log\left|\right(x-\frac{3}{2})+\sqrt{x^2-3x+2}|+C$