What is the integral of \sqrt[3]{x}(x+4) ?

The integral of \sqrt[3]{x}(x+4) is 3/7 x^{7/3}+3x^{4/3}+C

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integral of \sqrt[3]{x}(x+4)

\int 3 x (x + 4) d x

\frac{3}{7} x^{\frac{7}{3}} + 3 x^{\frac{4}{3}} + C

Solution steps

\int 3 x (x + 4) d x

Expand

3 x (x + 4) : x^{\frac{4}{3}} + 4 3 x

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

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

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

= \frac{3}{7} x^{\frac{7}{3}} + 3 x^{\frac{4}{3}}

Add a constant to the solution

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

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What is the integral of \sqrt[3]{x}(x+4) ?
The integral of \sqrt[3]{x}(x+4) is 3/7 x^{7/3}+3x^{4/3}+C