What is the integral of 3x^2(x^3+8)^3 ?

The integral of 3x^2(x^3+8)^3 is (x^{12})/4+8x^9+96x^6+512x^3+C

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integral of 3x^2(x^3+8)^3

\int 3 x^{2} (x^{3} + 8)^{3} d x

\frac{x ^{12}}{4} + 8 x^{9} + 96 x^{6} + 512 x^{3} + C

Solution steps

\int 3 x^{2} (x^{3} + 8)^{3} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int x^{2} (x^{3} + 8)^{3} d x

Expand

x^{2} (x^{3} + 8)^{3} : x^{11} + 24 x^{8} + 192 x^{5} + 512 x^{2}

= 3 \cdot \int x^{11} + 24 x^{8} + 192 x^{5} + 512 x^{2} 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

= 3 (\int x^{11} d x + \int 24 x^{8} d x + \int 192 x^{5} d x + \int 512 x^{2} d x)

\int x^{11} d x = \frac{x ^{12}}{12}

\int 24 x^{8} d x = \frac{8 x ^{9}}{3}

\int 192 x^{5} d x = 32 x^{6}

\int 512 x^{2} d x = \frac{512 x ^{3}}{3}

= 3 (\frac{x ^{12}}{12} + \frac{8 x ^{9}}{3} + 32 x^{6} + \frac{512 x ^{3}}{3})

Simplify

3 (\frac{x ^{12}}{12} + \frac{8 x ^{9}}{3} + 32 x^{6} + \frac{512 x ^{3}}{3}) : \frac{x ^{12}}{4} + 8 x^{9} + 96 x^{6} + 512 x^{3}

= \frac{x ^{12}}{4} + 8 x^{9} + 96 x^{6} + 512 x^{3}

Add a constant to the solution

= \frac{x ^{12}}{4} + 8 x^{9} + 96 x^{6} + 512 x^{3} + C

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What is the integral of 3x^2(x^3+8)^3 ?
The integral of 3x^2(x^3+8)^3 is (x^{12})/4+8x^9+96x^6+512x^3+C