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

The integral of (x^2+\sqrt[3]{x})^2 is (x^5)/5+3/5 x^{10/3}+3/5 x^{5/3}+C

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

\frac{x ^{5}}{5} + \frac{3}{5} x^{\frac{10}{3}} + \frac{3}{5} x^{\frac{5}{3}} + C

Solution steps

Expand

= \int x^{4} + 2 x^{\frac{7}{3}} + x^{\frac{2}{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 x^{4} d x + \int 2 x^{\frac{7}{3}} d x + \int x^{\frac{2}{3}} d x

\int x^{4} d x = \frac{x ^{5}}{5}

\int 2 x^{\frac{7}{3}} d x = \frac{3}{5} x^{\frac{10}{3}}

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

= \frac{x ^{5}}{5} + \frac{3}{5} x^{\frac{10}{3}} + \frac{3}{5} x^{\frac{5}{3}}

Add a constant to the solution

= \frac{x ^{5}}{5} + \frac{3}{5} x^{\frac{10}{3}} + \frac{3}{5} x^{\frac{5}{3}} + C

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What is the integral of (x^2+\sqrt[3]{x})^2 ?
The integral of (x^2+\sqrt[3]{x})^2 is (x^5)/5+3/5 x^{10/3}+3/5 x^{5/3}+C