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

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

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integral of (2x^2+x)2

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

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

Solution steps

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

Take the constant out:

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

= 2 \cdot \int 2 x^{2} + 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

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

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

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

= 2 (\frac{2 x ^{3}}{3} + \frac{x ^{2}}{2})

Simplify

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

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

Add a constant to the solution

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

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