What is the integral of x^{1/3}(2-x)^2 ?

The integral of x^{1/3}(2-x)^2 is 3x^{4/3}-12/7 x^{7/3}+3/10 x^{10/3}+C

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integral of x^{1/3}(2-x)^2

\int x^{\frac{1}{3}} (2 - x)^{2} d x

3 x^{\frac{4}{3}} - \frac{12}{7} x^{\frac{7}{3}} + \frac{3}{10} x^{\frac{10}{3}} + C

Solution steps

\int x^{\frac{1}{3}} (2 - x)^{2} d x

Expand

x^{\frac{1}{3}} (2 - x)^{2} : 4 x^{\frac{1}{3}} - 4 x^{\frac{4}{3}} + x^{\frac{7}{3}}

= \int 4 x^{\frac{1}{3}} - 4 x^{\frac{4}{3}} + x^{\frac{7}{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 4 x^{\frac{1}{3}} d x - \int 4 x^{\frac{4}{3}} d x + \int x^{\frac{7}{3}} d x

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

\int 4 x^{\frac{4}{3}} d x = \frac{12}{7} x^{\frac{7}{3}}

\int x^{\frac{7}{3}} d x = \frac{3}{10} x^{\frac{10}{3}}

= 3 x^{\frac{4}{3}} - \frac{12}{7} x^{\frac{7}{3}} + \frac{3}{10} x^{\frac{10}{3}}

Add a constant to the solution

= 3 x^{\frac{4}{3}} - \frac{12}{7} x^{\frac{7}{3}} + \frac{3}{10} x^{\frac{10}{3}} + C

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What is the integral of x^{1/3}(2-x)^2 ?
The integral of x^{1/3}(2-x)^2 is 3x^{4/3}-12/7 x^{7/3}+3/10 x^{10/3}+C