What is the integral from 0 to 1 of x^9*e^{-x} ?

The integral from 0 to 1 of x^9*e^{-x} is 362880-(986410)/e

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integral from 0 to 1 of x^9*e^{-x}

\int_{0}^{1} x^{9} \cdot e^{- x} d x

362880 - \frac{986410}{e}

Decimal

0.04043 \dots

Solution steps

\int_{0}^{1} x^{9} e^{- x} d x

Apply u-substitution

= \int_{0}^{- 1} e^{u} u^{9} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- 1}^{0} e^{u} u^{9} d u

Apply Integration By Parts

= - [u^{9} e^{u} - \int 9 u^{8} e^{u} d u]_{- 1}^{0}

\int 9 u^{8} e^{u} d u = 9 (u^{8} e^{u} - 8 (e^{u} u^{7} - 7 (e^{u} u^{6} - 6 (e^{u} u^{5} - 5 (e^{u} u^{4} - 4 (e^{u} u^{3} - 3 (e^{u} u^{2} - 2 (e^{u} u - e^{u}))))))))

= - [u^{9} e^{u} - 9 (u^{8} e^{u} - 8 (e^{u} u^{7} - 7 (e^{u} u^{6} - 6 (e^{u} u^{5} - 5 (e^{u} u^{4} - 4 (e^{u} u^{3} - 3 (e^{u} u^{2} - 2 (e^{u} u - e^{u}))))))))]_{- 1}^{0}

Compute the boundaries:

- 362880 + \frac{986410}{e}

= - (- 362880 + \frac{986410}{e})

Simplify

= 362880 - \frac{986410}{e}

What is the integral from 0 to 1 of x^9*e^{-x} ?
The integral from 0 to 1 of x^9*e^{-x} is 362880-(986410)/e