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

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

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integral of x(x+3)^2

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

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

Solution steps

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

Expand

x (x + 3)^{2} : x^{3} + 6 x^{2} + 9 x

= \int x^{3} + 6 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 x^{3} d x + \int 6 x^{2} d x + \int 9 x d x

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

\int 6 x^{2} d x = 2 x^{3}

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

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

Add a constant to the solution

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

(3 x^{2} + 3 x - 1)^{4}

10 \frac{d ^{2} y}{d x ^{2}} + 3 \frac{d y}{d x} + 6 y = 0

n = 2 \sum \infty \frac{2}{n ^{2} + 4 n + 3}

\frac{1}{∣ 1 - x ∣}

\frac{d y}{d x} = \frac{3 ( 2 + y )}{( 3 - x ) ^{2}}, y (0) = - 1

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