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

The integral of (x-2)(x+3)^2 is (x^4)/4+(4x^3)/3-(3x^2)/2-18x+C

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

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

\frac{x ^{4}}{4} + \frac{4 x ^{3}}{3} - \frac{3 x ^{2}}{2} - 18 x + C

Solution steps

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

Expand

(x - 2) (x + 3)^{2} : x^{3} + 4 x^{2} - 3 x - 18

= \int x^{3} + 4 x^{2} - 3 x - 18 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 4 x^{2} d x - \int 3 x d x - \int 18 d x

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

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

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

\int 18 d x = 18 x

= \frac{x ^{4}}{4} + \frac{4 x ^{3}}{3} - \frac{3 x ^{2}}{2} - 18 x

Add a constant to the solution

= \frac{x ^{4}}{4} + \frac{4 x ^{3}}{3} - \frac{3 x ^{2}}{2} - 18 x + C

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What is the integral of (x-2)(x+3)^2 ?
The integral of (x-2)(x+3)^2 is (x^4)/4+(4x^3)/3-(3x^2)/2-18x+C