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

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

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integral of x(2x+3)^2

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

x^{4} + 4 x^{3} + \frac{9 x ^{2}}{2} + C

Solution steps

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

Expand

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

= \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

= \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}

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

Add a constant to the solution

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

\frac{d y}{d x} = \frac{y cos ( x )}{1 - 2 y}

\frac{d}{d x} (12)

\frac{d}{d x} (x ln (cos (x)))

x \to + 0 lim (\frac{2 ( y + x ) - 2 y}{x})

x \to \infty lim (\frac{x}{x})

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