Question

int_{9}^{4} (sqrt x+frac{1}{sqrt x})^{2}dx

Integrals
ANSWERED
asked 2021-09-11

\(\int_{9}^{4} \bigg(\sqrt x+\frac{1}{\sqrt x}\bigg)^{2}dx\)

Expert Answers (1)

2021-09-12

Compute the definite integral:
For the integrand \(\displaystyle{\left(\sqrt{{{x}}}+\frac{1}{\sqrt{{{x}}}}\right)}^{2}\), do long division:
\(\displaystyle={\int_{{4}}^{{9}}}{x}+\frac{1}{{x}}+{2}{\left.{d}{x}\right.}\)
Integrate the sum term by term and factor out constants:
\(\displaystyle={\int_{{4}}^{{9}}}{x}{\left.{d}{x}\right.}+{\int_{{4}}^{{9}}}\frac{1}{{x}}{\left.{d}{x}\right.}+{2}{\int_{{4}}^{{9}}}{1}{\left.{d}{x}\right.}\)
Apply the fundamental theorem of calculus.
The antiderivative of x is \(\displaystyle\frac{{x}^{2}}{{2}}\):
\(\displaystyle=\frac{{x}^{2}}{{2}}\) right bracketing bar \(\displaystyle_{4}^{9}+\int{}{}{}{}{{}_{{4}}^{{9}}}\frac{1}{{x}}{\left.{d}{x}\right.}+{2}\int{}{}{}{}{{}_{{4}}^{{9}}}{1}{\left.{d}{x}\right.}\)
Evaluate the antiderivative at the limits and subtract.
\(\displaystyle\frac{{x}^{2}}{{2}}\) right bracketing bar \(\displaystyle_{4}^{9}=\frac{{9}^{2}}{{2}}-\frac{{4}^{2}}{{2}}=\frac{65}{{2}}:\)
\(\displaystyle=\frac{65}{{2}}+{\int_{{4}}^{{9}}}\frac{1}{{x}}{\left.{d}{x}\right.}+{2}{\int_{{4}}^{{9}}}{1}{\left.{d}{x}\right.}\)
Apply the fundamental theorem of calculus.
The antiderivative of \(\displaystyle\frac{1}{{x}}\) is \(\displaystyle \log{{\left({x}\right)}}\):
\(\displaystyle=\frac{65}{{2}}+ \log{{\left({x}\right)}}\) right bracketing bar \(\displaystyle_{4}^{9}+{2}\int{}{}{}{}{{}_{{4}}^{{9}}}{1}{\left.{d}{x}\right.}\)
Evaluate the antiderivative at the limits and subtract.
\(\displaystyle \log{{\left({x}\right)}}\) right bracketing bar \(\displaystyle_{4}^{9}= \log{{\left({9}\right)}}- \log{{\left({4}\right)}}= \log{{\left(\frac{9}{{4}}\right)}}\):
\(\displaystyle=\frac{65}{{2}}+ \log{{\left(\frac{9}{{4}}\right)}}+{2}{\int_{{4}}^{{9}}}{1}{\left.{d}{x}\right.}\)
Apply the fundamental theorem of calculus.
The antiderivative of 1 is x:
\(\displaystyle=\frac{65}{{2}}+ \log{{\left(\frac{9}{{4}}\right)}}+{2}{x}\) right bracketing bar \(\displaystyle_{4}^{9}\)
Evaluate the antiderivative at the limits and subtract.
2x right bracketing bar \(\displaystyle_{4}^{9}={2}{9}-{2}{4}={10}\):
Answer:
\(\displaystyle=\frac{85}{{2}}+ \log{{\left(\frac{9}{{4}}\right)}}\)

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