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

Lösung

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

362880 - \frac{986410}{e}

Dezimale

0.04043 \dots

Schritte zur Lösung

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

Wende U-Substitution an

= \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

Wende die partielle Integration an

= - [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}

Berechne die Grenzen:

- 362880 + \frac{986410}{e}

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

Vereinfache

= 362880 - \frac{986410}{e}

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