Question

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Answer 1

$\displaystyle \bold \red{\int{\left(\sqrt[3]{x} + \frac{1}{\sqrt{x}}\right)d x}}$

Integrate term by term:

${ \displaystyle{ \bold \red{\color{red}{\int{\left(\sqrt[3]{x} + \frac{1}{\sqrt{x}}\right)d x}} = \color{red}{\left(\int{\frac{1}{\sqrt{x}} d x} + \int{\sqrt[3]{x} d x}\right)}}}}$

Apply the power rule

$\displaystyle{ \bold \red{\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1} \left(n \neq -1 \right) \: with \: n=- \frac{1}{2}}}$

${ \displaystyle{\bold \red{\int{\sqrt[3]{x} d x} + \color{red}{\int{\frac{1}{\sqrt{x}} d x}}=\int{\sqrt[3]{x} d x} + \color{red}{\int{x^{- \frac{1}{2}} d x}}=\int{\sqrt[3]{x} d x} + \color{red}{\frac{x^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}}$

$\displaystyle{\bold \red{=\int{\sqrt[3]{x} d x} + \color{red}{\left(2 x^{\frac{1}{2}}\right)}=\int{\sqrt[3]{x} d x} + \color{red}{\left(2 \sqrt{x}\right)}}}$

Apply the power rule

${ \displaystyle \bold \red{\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1} \left(n \neq -1 \right) \: with \: n=\frac{1}{3}}}$

$\displaystyle{ \bold \red{2 \sqrt{x} + \color{red}{\int{\sqrt[3]{x} d x}}=2 \sqrt{x} + \color{red}{\int{x^{\frac{1}{3}} d x}}=2 \sqrt{x} + \color{red}{\frac{x^{\frac{1}{3} + 1}}{\frac{1}{3} + 1}}}}$

$\displaystyle \bold \red{=2 \sqrt{x} + \color{red}{\left(\frac{3 x^{\frac{4}{3}}}{4}\right)}}$

Therefore,

${ \displaystyle \bold \red{\int{\left(\sqrt[3]{x} + \frac{1}{\sqrt{x}}\right)d x} = \frac{3 x^{\frac{4}{3}}}{4} + 2 \sqrt{x}}}$

Add the constant of integration:

$\displaystyle{\bold \red{\int{\left(\sqrt[3]{x} + \frac{1}{\sqrt{x}}\right)d x} = \frac{3 x^{\frac{4}{3}}}{4} + 2 \sqrt{x}+C}}$