What is the integral of ((x-3)^2)/(\sqrt[3]{x)} ?

The integral of ((x-3)^2)/(\sqrt[3]{x)} is 3/8 x^{8/3}-18/5 x^{5/3}+27/2 x^{2/3}+C

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integral of ((x-3)^2)/(\sqrt[3]{x)}

\int \frac{( x - 3 ) ^{2}}{3 x} d x

\frac{3}{8} x^{\frac{8}{3}} - \frac{18}{5} x^{\frac{5}{3}} + \frac{27}{2} x^{\frac{2}{3}} + C

Solution steps

\int \frac{( x - 3 ) ^{2}}{3 x} d x

Expand

\frac{( x - 3 ) ^{2}}{3 x} : x^{\frac{5}{3}} - 6 x^{\frac{2}{3}} + \frac{9}{3 x}

= \int x^{\frac{5}{3}} - 6 x^{\frac{2}{3}} + \frac{9}{3 x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{\frac{5}{3}} d x - \int 6 x^{\frac{2}{3}} d x + \int \frac{9}{3 x} d x

\int x^{\frac{5}{3}} d x = \frac{3}{8} x^{\frac{8}{3}}

\int 6 x^{\frac{2}{3}} d x = \frac{18}{5} x^{\frac{5}{3}}

\int \frac{9}{3 x} d x = \frac{27}{2} x^{\frac{2}{3}}

= \frac{3}{8} x^{\frac{8}{3}} - \frac{18}{5} x^{\frac{5}{3}} + \frac{27}{2} x^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{8} x^{\frac{8}{3}} - \frac{18}{5} x^{\frac{5}{3}} + \frac{27}{2} x^{\frac{2}{3}} + C

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What is the integral of ((x-3)^2)/(\sqrt[3]{x)} ?
The integral of ((x-3)^2)/(\sqrt[3]{x)} is 3/8 x^{8/3}-18/5 x^{5/3}+27/2 x^{2/3}+C