Evaluate the following integrals. int(x/(sqrt(x-4))dx

Question
Applications of integrals
asked 2021-01-13
Evaluate the following integrals.
\(\displaystyle\int{\left(\frac{{x}}{{\sqrt{{{x}-{4}}}}}{\left.{d}{x}\right.}\right.}\)

Answers (1)

2021-01-14
Step 1
Consider the integrals,
\(\displaystyle\int{\left(\frac{{x}}{{\sqrt{{{x}-{4}}}}}{\left.{d}{x}\right.}\right.}\)
Suppose that, \(\displaystyle\sqrt{{{x}-{4}}}={t}\)
Differentiating with respect to "x"
\(\displaystyle\frac{{1}}{{{2}\sqrt{{{x}-{4}}}}}{\left.{d}{x}\right.}={\left.{d}{t}\right.}\)
\(\displaystyle\frac{{1}}{{\sqrt{{{x}-{4}}}}}{\left.{d}{x}\right.}={2}{\left.{d}{t}\right.}\)
Step 2
Substitute all value in given integrals,
\(\displaystyle\int\frac{{x}}{{\sqrt{{{x}-{4}}}}}{\left.{d}{x}\right.}=\int{\left({t}^{{2}}+{4}\right)}{2}{\left.{d}{t}\right.}\)
\(\displaystyle={2}\int{\left({t}^{{2}}+{4}\right)}{\left.{d}{t}\right.}\)
\(\displaystyle={2}{\left[\frac{{t}^{{3}}}{{3}}+{4}{t}\right]}+{C}\)
\(\displaystyle=\frac{{{2}{t}^{{3}}}}{{3}}+{8}{t}+{C}\)
\(\displaystyle=\frac{{2}}{{3}}{\left({x}-{4}\right)}^{{\frac{{3}}{{2}}}}+{8}\sqrt{{{x}-{4}}}+{C}\)
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