# Evaluate the following integrals. \int \frac{dx}{\sqrt{(x-1)(3-x)}}

Tabansi 2020-10-23 Answered
Evaluate the following integrals.
$\int \frac{dx}{\sqrt{\left(x-1\right)\left(3-x\right)}}$
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## Expert Answer

Neelam Wainwright
Answered 2020-10-24 Author has 102 answers
Step 1: Given that
$\int \frac{dx}{\sqrt{\left(x-1\right)\left(3-x\right)}}$
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,
$\int \frac{dx}{\sqrt{\left(x-1\right)\left(3-x\right)}}=\int \frac{dx}{\sqrt{3x-{x}^{2}-3+x}}$
$=\int \frac{dx}{\sqrt{-{x}^{2}+4x-3}}$
$=\int \frac{dx}{\sqrt{-\left({x}^{2}-4x+3\right)}}$
$=\int \frac{dx}{\sqrt{-\left({x}^{2}-4x+{2}^{2}-{2}^{2}+3\right)}}$
$=\int \frac{dx}{\sqrt{-\left({\left(x-2\right)}^{2}-4+3\right)}}$
$=\int \frac{dx}{\sqrt{-\left({\left(x-2\right)}^{2}-1\right)}}$
$=\int \frac{dx}{\sqrt{1-{\left(x-2\right)}^{2}}}$
$={\mathrm{sin}}^{-1}\left(x-2\right)+C$

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