What is the integral from 0 to 1 of (x-x^2)x ?

The integral from 0 to 1 of (x-x^2)x is 1/12

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integral from 0 to 1 of (x-x^2)x

\int_{0}^{1} (x - x^{2}) x d x

\frac{1}{12}

Solution steps

\int_{0}^{1} (x - x^{2}) x d x

Expand

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

= \int_{0}^{1} 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

= \int_{0}^{1} x^{2} d x - \int_{0}^{1} x^{3} d x

\int_{0}^{1} x^{2} d x = \frac{1}{3}

\int_{0}^{1} x^{3} d x = \frac{1}{4}

= \frac{1}{3} - \frac{1}{4}

Simplify

\frac{1}{3} - \frac{1}{4} : \frac{1}{12}

= \frac{1}{12}

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What is the integral from 0 to 1 of (x-x^2)x ?
The integral from 0 to 1 of (x-x^2)x is 1/12