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

Question
Applications of integrals
Evaluate the following integrals.
$$\displaystyle\int{\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{\left({x}-{1}\right)}{\left({3}-{x}\right)}}}}}}$$

2020-10-24
Step 1: Given that
$$\displaystyle\int{\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{\left({x}-{1}\right)}{\left({3}-{x}\right)}}}}}}$$
Step 2: Formula Used
$$\displaystyle\int{\frac{{{1}}}{{\sqrt{{{a}^{{{2}}}-{x}^{{{2}}}}}}}}{\left.{d}{x}\right.}={{\sin}^{{-{1}}}{\left({\frac{{{x}}}{{{a}}}}\right)}}+{C}$$
Step 3: Solve
We have,
$$\displaystyle\int{\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{\left({x}-{1}\right)}{\left({3}-{x}\right)}}}}}}=\int{\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{3}{x}-{x}^{{{2}}}-{3}+{x}}}}}}$$
$$\displaystyle=\int{\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{x}^{{{2}}}+{4}{x}-{3}}}}}}$$
$$\displaystyle=\int{\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{\left({x}^{{{2}}}-{4}{x}+{3}\right)}}}}}}$$
$$\displaystyle=\int{\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{\left({x}^{{{2}}}-{4}{x}+{2}^{{{2}}}-{2}^{{{2}}}+{3}\right)}}}}}}$$
$$\displaystyle=\int{\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{\left({\left({x}-{2}\right)}^{{{2}}}-{4}+{3}\right)}}}}}}$$
$$\displaystyle=\int{\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{\left({\left({x}-{2}\right)}^{{{2}}}-{1}\right)}}}}}}$$
$$\displaystyle=\int{\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{1}-{\left({x}-{2}\right)}^{{{2}}}}}}}}$$
$$\displaystyle={{\sin}^{{-{1}}}{\left({x}-{2}\right)}}+{C}$$

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