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人気のある微分積分 >

integral from 0 to 1 of x^5e^{-x}

解

\int_{0}^{1} x^{5} e^{- x} d x

解

120 - \frac{326}{e}

+1

十進法表記

0.07130 \dots

解答ステップ

\int_{0}^{1} x^{5} e^{- x} d x

u置換積分法を適用する

= \int_{0}^{- 1} e^{u} u^{5} d u

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

= - \int_{- 1}^{0} e^{u} u^{5} d u

部分積分を適用する

= - [u^{5} e^{u} - \int 5 u^{4} e^{u} d u]_{- 1}^{0}

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

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

限度を計算する:

- 120 + \frac{326}{e}

= - (- 120 + \frac{326}{e})

簡素化

= 120 - \frac{326}{e}

グラフ

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