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

解答

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

120 - \frac{326}{e}

十进制

0.07130 \dots

求解步骤

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

使用换元积分法

= \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

使用分布积分法

= - [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}

带入上下限计算后得:

- 120 + \frac{326}{e}

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

化简

= 120 - \frac{326}{e}

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