integral from 0 to 1 of xe^{-6x}

解答

\int_{0}^{1} x e^{- 6 x} d x

- \frac{- e ^{6} + 7}{36 e ^{6}}

十进制

0.02729 \dots

求解步骤

\int_{0}^{1} x e^{- 6 x} d x

使用换元积分法

= \int_{0}^{- 6} \frac{e ^{u} u}{36} d u

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

= - \int_{- 6}^{0} \frac{e ^{u} u}{36} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{36} \cdot \int_{- 6}^{0} e^{u} u d u

使用分布积分法

= - \frac{1}{36} [e^{u} u - \int e^{u} d u]_{- 6}^{0}

\int e^{u} d u = e^{u}

= - \frac{1}{36} [e^{u} u - e^{u}]_{- 6}^{0}

带入上下限计算后得:

- 1 + \frac{7}{e ^{6}}

= - \frac{1}{36} (- 1 + \frac{7}{e ^{6}})

化简

= - \frac{- e ^{6} + 7}{36 e ^{6}}

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