Question

Evaluate

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

Answer 1

We have,

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

Now,

$\int \frac{dx}{\sqrt{x^2-x-2x+2}}$

$\Rightarrow \int \frac{dx}{x^2-3x+2}$

Making completing square,

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

$\Rightarrow \int \frac{dx}{\sqrt{x^2-3x+\frac{9}{4}-\frac{9}{4}+2}}$

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

$\Rightarrow \int \frac{dx}{\sqrt{{(x-\frac{3}{4})}^2-{\left(\frac{1}{2}\right)}^2}}$

Applying formula,

$\int \frac{1}{\sqrt{x^2-a^2}}dx=\ln |x+\sqrt{x^2-a^2}|$

So,

$\int \frac{dx}{\sqrt{{(x-\frac{3}{4})}^2-{\left(\frac{1}{2}\right)}^2}}=\ln |(x-\frac{3}{4})+\sqrt{{(x-\frac{3}{4})}^2-{\left(\frac{1}{2}\right)}^2}|$

$=\ln |(x-\frac{3}{4})+\sqrt{x^2-3x+2}|$

Hence, this is the answer