What is the integral from 0 to 1 of 2x-x^2-x^3 ?

The integral from 0 to 1 of 2x-x^2-x^3 is 5/12

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integral from 0 to 1 of 2x-x^2-x^3

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

\frac{5}{12}

Decimal

0.41666 \dots

Solution steps

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

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

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

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

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

Simplify

1 - \frac{1}{3} - \frac{1}{4} : \frac{5}{12}

= \frac{5}{12}

What is the integral from 0 to 1 of 2x-x^2-x^3 ?
The integral from 0 to 1 of 2x-x^2-x^3 is 5/12