# Evaluate the following integrals. int(dx)/(sqrt((x-1)(3-x)))

Evaluate the following integrals.
$\int \frac{dx}{\sqrt{\left(x-1\right)\left(3-x\right)}}$
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Step 1: Given that
Completing the square Evaluate the following integrals.
$\int \frac{dx}{\sqrt{\left(x-1\right)\left(3-x\right)}}$
Step 2: Formula Used
$\int \frac{1}{\sqrt{{a}^{2}-{x}^{2}}}dx={\mathrm{sin}}^{-1}\left(\frac{x}{a}\right)+C$
Step 3:Solve
We have,
$\int \frac{dx}{\sqrt{\left(x-1\right)\left(3-x\right)}}=\int \frac{dx}{\sqrt{3x-{x}^{2}-3+x}}$
$=\int \frac{dx}{\sqrt{-{x}^{2}+4x-3}}$
$=\int \frac{dx}{\sqrt{-\left({x}^{2}-4x+3\right)}}$
$=\int \frac{dx}{\sqrt{-\left({x}^{2}-4x+{2}^{2}-{2}^{2}+3\right)}}$
$=\int \frac{dx}{\sqrt{-{\left(x-2\right)}^{2}-4+3}}$
$=\int \frac{dx}{\sqrt{-\left({\left(x-2\right)}^{2}\right)-1}}$
$=\int \frac{dx}{\sqrt{1-{\left(x-2\right)}^{2}}}$
$={\mathrm{sin}}^{-1}\left(x-2\right)+C$