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

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

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integral of-(x-2)^2

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

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

Solution steps

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

Take the constant out:

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

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

Expand

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

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

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

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

\int 4 d x = 4 x

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

Simplify

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

Add a constant to the solution

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

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