Question

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

Answer 1

To find $\int \frac{1}{\sqrt{(x-1)(x-2)}}.dx$

$=\int \frac{1}{\sqrt{x^2-3x+2}}.dx$

$=\int \frac{1}{\sqrt{x^2-3x+{\left(\frac{3}{2}\right)}^2-{\left(\frac{3}{2}\right)}^2+2}}.dx$

$=\int \frac{1}{\sqrt{{(x-\frac{3}{2})}^2-\frac{9}{4}+2}}.dx$

$=\int \frac{1}{\sqrt{{(x-\frac{3}{2})}^2-\frac{1}{4}}}.dx$

$=\int \frac{1}{\sqrt{{(x-\frac{3}{2})}^2-{\left(\frac{1}{2}\right)}^2}}.dx$

$=\text{log}|x-\frac{3}{2}+\sqrt{{(x-\frac{3}{2})}^2-{\left(\frac{1}{2}\right)}^2}|+C$

$[\because \int \frac{dx}{\sqrt{x^2-a^2}}=\text{log}|x+\sqrt{x^2-a^2}|+C]$

$=\text{log}|x-\frac{3}{2}+\sqrt{x^2-3x+2}||+C$