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受欢迎的微积分 >

integral from 0 to 1 of x^2e^{-12x}

解答

\int_{0}^{1} x^{2} e^{- 12 x} d x

解答

\frac{2 e ^{12} - 170}{1728 e ^{12}}

+1

十进制

0.00115 \dots

求解步骤

\int_{0}^{1} x^{2} e^{- 12 x} d x

使用换元积分法

= \int_{0}^{- 12} - \frac{e ^{u} u ^{2}}{1728} d u

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

= - \int_{- 12}^{0} - \frac{e ^{u} u ^{2}}{1728} d u

提出常数:

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

= - (- \frac{1}{1728} \cdot \int_{- 12}^{0} e^{u} u^{2} d u)

使用分布积分法

= - (- \frac{1}{1728} [u^{2} e^{u} - \int 2 u e^{u} d u]_{- 12}^{0})

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

= - (- \frac{1}{1728} [u^{2} e^{u} - 2 (e^{u} u - e^{u})]_{- 12}^{0})

化简

= \frac{1}{1728} [e^{u} u^{2} - 2 (e^{u} u - e^{u})]_{- 12}^{0}

带入上下限计算后得:

2 - \frac{170}{e ^{12}}

= \frac{1}{1728} (2 - \frac{170}{e ^{12}})

化简

= \frac{2 e ^{12} - 170}{1728 e ^{12}}

作图

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