Step 1: Given that
Completing the square Evaluate the following integrals.
\(\displaystyle\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{\left({x}-{1}\right)}{\left({3}-{x}\right)}}}}}\)
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,
\(\displaystyle\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{\left({x}-{1}\right)}{\left({3}-{x}\right)}}}}}=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{3}{x}-{x}^{{2}}-{3}+{x}}}}}\)
\(\displaystyle=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{x}^{{2}}+{4}{x}-{3}}}}}\)
\(\displaystyle=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{\left({x}^{{2}}-{4}{x}+{3}\right)}}}}}\)
\(\displaystyle=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{\left({x}^{{2}}-{4}{x}+{2}^{{2}}-{2}^{{2}}+{3}\right)}}}}}\)
\(\displaystyle=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{\left({x}-{2}\right)}^{{2}}-{4}+{3}}}}}\)
\(\displaystyle=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{\left({\left({x}-{2}\right)}^{{2}}\right)}-{1}}}}}\)
\(\displaystyle=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{1}-{\left({x}-{2}\right)}^{{2}}}}}}\)
\(\displaystyle={{\sin}^{{-{{1}}}}{\left({x}-{2}\right)}}+{C}\)
Completing the square Evaluate the following integrals.
\(\displaystyle\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{\left({x}-{1}\right)}{\left({3}-{x}\right)}}}}}\)
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,
\(\displaystyle\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{\left({x}-{1}\right)}{\left({3}-{x}\right)}}}}}=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{3}{x}-{x}^{{2}}-{3}+{x}}}}}\)
\(\displaystyle=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{x}^{{2}}+{4}{x}-{3}}}}}\)
\(\displaystyle=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{\left({x}^{{2}}-{4}{x}+{3}\right)}}}}}\)
\(\displaystyle=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{\left({x}^{{2}}-{4}{x}+{2}^{{2}}-{2}^{{2}}+{3}\right)}}}}}\)
\(\displaystyle=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{\left({x}-{2}\right)}^{{2}}-{4}+{3}}}}}\)
\(\displaystyle=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{-{\left({\left({x}-{2}\right)}^{{2}}\right)}-{1}}}}}\)
\(\displaystyle=\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{1}-{\left({x}-{2}\right)}^{{2}}}}}}\)
\(\displaystyle={{\sin}^{{-{{1}}}}{\left({x}-{2}\right)}}+{C}\)
