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

The integral of 3x^2(x^3+4) is (x^6)/2+4x^3+C

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

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

\frac{x ^{6}}{2} + 4 x^{3} + C

Solution steps

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

Expand

x^{2} (x^{3} + 4) : x^{5} + 4 x^{2}

= 3 \cdot \int x^{5} + 4 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^{5} d x + \int 4 x^{2} d x)

\int x^{5} d x = \frac{x ^{6}}{6}

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

= 3 (\frac{x ^{6}}{6} + \frac{4 x ^{3}}{3})

Simplify

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

= \frac{x ^{6}}{2} + 4 x^{3}

Add a constant to the solution

= \frac{x ^{6}}{2} + 4 x^{3} + C

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What is the integral of 3x^2(x^3+4) ?
The integral of 3x^2(x^3+4) is (x^6)/2+4x^3+C