Question

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

Answer 1

$(x-a)(x-b)=x^2-(a+b)x+ab=x^2-(a+b)x+\frac{(a+b{)}^2}{4}-\frac{(a+b{)}^2}{4}+ab$
$={[x-\left(\frac{a+b}{2}\right)]}^2-\frac{(a-b{)}^2}{4}$
$\Rightarrow \int \frac{1}{\sqrt{(x-a)(x-b)}}dx=\int \frac{1}{\sqrt{{\{x-\left(\frac{a+b}{2}\right)\}}^2-{\left(\frac{a-b}{2}\right)}^2}}dx$
Let $x-\left(\frac{a+b}{2}\right)=t\Rightarrow dx=dt$
$\Rightarrow \int \frac{1}{\sqrt{{\{x-\left(\frac{a+b}{2}\right)\}}^2-{\left(\frac{a-b}{2}\right)}^2}}dx=\int \frac{1}{\sqrt{t^2-{\left(\frac{a-b}{2}\right)}^2}}dt$
$=\log |t+\sqrt{t^2-{\left(\frac{a-b}{2}\right)}^2}|+C$
$=\log |\{x-\left(\frac{a+b}{2}\right)\}+\sqrt{(x-a)(x-b)}|+C$