What is the integral of 12x^2(x^3+4)^3 ?

The integral of 12x^2(x^3+4)^3 is x^{12}+16x^9+96x^6+256x^3+C

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integral of 12x^2(x^3+4)^3

\int 12 x^{2} (x^{3} + 4)^{3} d x

x^{12} + 16 x^{9} + 96 x^{6} + 256 x^{3} + C

Solution steps

\int 12 x^{2} (x^{3} + 4)^{3} d x

Take the constant out:

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

= 12 \cdot \int x^{2} (x^{3} + 4)^{3} d x

Expand

x^{2} (x^{3} + 4)^{3} : x^{11} + 12 x^{8} + 48 x^{5} + 64 x^{2}

= 12 \cdot \int x^{11} + 12 x^{8} + 48 x^{5} + 64 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

= 12 (\int x^{11} d x + \int 12 x^{8} d x + \int 48 x^{5} d x + \int 64 x^{2} d x)

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

\int 12 x^{8} d x = \frac{4 x ^{9}}{3}

\int 48 x^{5} d x = 8 x^{6}

\int 64 x^{2} d x = \frac{64 x ^{3}}{3}

= 12 (\frac{x ^{12}}{12} + \frac{4 x ^{9}}{3} + 8 x^{6} + \frac{64 x ^{3}}{3})

Simplify

12 (\frac{x ^{12}}{12} + \frac{4 x ^{9}}{3} + 8 x^{6} + \frac{64 x ^{3}}{3}) : x^{12} + 16 x^{9} + 96 x^{6} + 256 x^{3}

= x^{12} + 16 x^{9} + 96 x^{6} + 256 x^{3}

Add a constant to the solution

= x^{12} + 16 x^{9} + 96 x^{6} + 256 x^{3} + C

\frac{d}{d x} (x 2)

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

x, x^{1}, 0, 1

n \to \infty lim (\frac{n}{4})

x \to 0 + lim (x^{2} ln (3 x))

What is the integral of 12x^2(x^3+4)^3 ?
The integral of 12x^2(x^3+4)^3 is x^{12}+16x^9+96x^6+256x^3+C