What is the integral from 0 to 2 of xp(x-x^2) ?

The integral from 0 to 2 of xp(x-x^2) is -4/3 p

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integral from 0 to 2 of xp(x-x^2)

\int_{0}^{2} x p (x - x^{2}) d x

- \frac{4}{3} p

Solution steps

\int_{0}^{2} x p (x - x^{2}) d x

Take the constant out:

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

= p \cdot \int_{0}^{2} (x - x^{2}) x d x

Expand

(x - x^{2}) x : x^{2} - x^{3}

= p \cdot \int_{0}^{2} x^{2} - x^{3} 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

= p (\int_{0}^{2} x^{2} d x - \int_{0}^{2} x^{3} d x)

\int_{0}^{2} x^{2} d x = \frac{8}{3}

\int_{0}^{2} x^{3} d x = 4

= p (\frac{8}{3} - 4)

Simplify

p (\frac{8}{3} - 4) : - \frac{4}{3} p

= - \frac{4}{3} p

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What is the integral from 0 to 2 of xp(x-x^2) ?
The integral from 0 to 2 of xp(x-x^2) is -4/3 p