What is the integral of ((x^3)/3+2)^2x^3 ?

The integral of ((x^3)/3+2)^2x^3 is (x^{10})/(90)+(4x^7)/(21)+x^4+C

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integral of ((x^3)/3+2)^2x^3

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

\frac{x ^{10}}{90} + \frac{4 x ^{7}}{21} + x^{4} + C

Solution steps

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

Expand

(\frac{x ^{3}}{3} + 2)^{2} x^{3} : \frac{x ^{9}}{9} + \frac{4 x ^{6}}{3} + 4 x^{3}

= \int \frac{x ^{9}}{9} + \frac{4 x ^{6}}{3} + 4 x^{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{x ^{9}}{9} d x + \int \frac{4 x ^{6}}{3} d x + \int 4 x^{3} d x

\int \frac{x ^{9}}{9} d x = \frac{x ^{10}}{90}

\int \frac{4 x ^{6}}{3} d x = \frac{4 x ^{7}}{21}

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

= \frac{x ^{10}}{90} + \frac{4 x ^{7}}{21} + x^{4}

Add a constant to the solution

= \frac{x ^{10}}{90} + \frac{4 x ^{7}}{21} + x^{4} + C

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What is the integral of ((x^3)/3+2)^2x^3 ?
The integral of ((x^3)/3+2)^2x^3 is (x^{10})/(90)+(4x^7)/(21)+x^4+C