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

The integral of 2x(2x+3)^2 is 2x^4+8x^3+9x^2+C

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

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

2 x^{4} + 8 x^{3} + 9 x^{2} + C

Solution steps

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

Take the constant out:

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

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

Expand

x (2 x + 3)^{2} : 4 x^{3} + 12 x^{2} + 9 x

= 2 \cdot \int 4 x^{3} + 12 x^{2} + 9 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 4 x^{3} d x + \int 12 x^{2} d x + \int 9 x d x)

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

\int 12 x^{2} d x = 4 x^{3}

\int 9 x d x = \frac{9 x ^{2}}{2}

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

Simplify

2 (x^{4} + 4 x^{3} + \frac{9 x ^{2}}{2}) : 2 x^{4} + 8 x^{3} + 9 x^{2}

= 2 x^{4} + 8 x^{3} + 9 x^{2}

Add a constant to the solution

= 2 x^{4} + 8 x^{3} + 9 x^{2} + C

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What is the integral of 2x(2x+3)^2 ?
The integral of 2x(2x+3)^2 is 2x^4+8x^3+9x^2+C