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

The integral from 0 to 1 of (2x-x^2)-x^2 is 1/3

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

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

\frac{1}{3}

Solution steps

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

Simplify

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

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

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

= 1 - \frac{2}{3}

Simplify

1 - \frac{2}{3} : \frac{1}{3}

= \frac{1}{3}

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What is the integral from 0 to 1 of (2x-x^2)-x^2 ?
The integral from 0 to 1 of (2x-x^2)-x^2 is 1/3