What is the integral of ((1-x)^3)/(x*\sqrt[3]{x)} ?

The integral of ((1-x)^3)/(x*\sqrt[3]{x)} is -3/(\sqrt[3]{x)}-9/2 x^{2/3}+9/5 x^{5/3}-3/8 x^{8/3}+C

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integral of ((1-x)^3)/(x*\sqrt[3]{x)}

\int \frac{( 1 - x ) ^{3}}{x \cdot 3 x} d x

- \frac{3}{3 x} - \frac{9}{2} x^{\frac{2}{3}} + \frac{9}{5} x^{\frac{5}{3}} - \frac{3}{8} x^{\frac{8}{3}} + C

Solution steps

\int \frac{( 1 - x ) ^{3}}{x 3 x} d x

Expand

\frac{( 1 - x ) ^{3}}{x 3 x} : \frac{1}{x ^{\frac{4}{3}}} - \frac{3}{x ^{\frac{1}{3}}} + 3 x^{\frac{2}{3}} - x^{\frac{5}{3}}

= \int \frac{1}{x ^{\frac{4}{3}}} - \frac{3}{x ^{\frac{1}{3}}} + 3 x^{\frac{2}{3}} - x^{\frac{5}{3}} 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 \frac{1}{x ^{\frac{4}{3}}} d x - \int \frac{3}{x ^{\frac{1}{3}}} d x + \int 3 x^{\frac{2}{3}} d x - \int x^{\frac{5}{3}} d x

\int \frac{1}{x ^{\frac{4}{3}}} d x = - \frac{3}{3 x}

\int \frac{3}{x ^{\frac{1}{3}}} d x = \frac{9}{2} x^{\frac{2}{3}}

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

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

= - \frac{3}{3 x} - \frac{9}{2} x^{\frac{2}{3}} + \frac{9}{5} x^{\frac{5}{3}} - \frac{3}{8} x^{\frac{8}{3}}

Add a constant to the solution

= - \frac{3}{3 x} - \frac{9}{2} x^{\frac{2}{3}} + \frac{9}{5} x^{\frac{5}{3}} - \frac{3}{8} x^{\frac{8}{3}} + C

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What is the integral of ((1-x)^3)/(x*\sqrt[3]{x)} ?
The integral of ((1-x)^3)/(x*\sqrt[3]{x)} is -3/(\sqrt[3]{x)}-9/2 x^{2/3}+9/5 x^{5/3}-3/8 x^{8/3}+C