What is the integral from 0 to 1 of xe^{-x}-x^2 ?

The integral from 0 to 1 of xe^{-x}-x^2 is 2/3-2/e

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integral from 0 to 1 of xe^{-x}-x^2

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

\frac{2}{3} - \frac{2}{e}

Decimal

- 0.06909 \dots

Solution steps

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

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

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

= - \frac{2}{e} + 1 - \frac{1}{3}

Simplify

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

= \frac{2}{3} - \frac{2}{e}

What is the integral from 0 to 1 of xe^{-x}-x^2 ?
The integral from 0 to 1 of xe^{-x}-x^2 is 2/3-2/e