Question

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

Integrals

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

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 $$\displaystyle_{4}^{9}={2}{9}-{2}{4}={10}$$:
$$\displaystyle=\frac{85}{{2}}+ \log{{\left(\frac{9}{{4}}\right)}}$$