integral from 0 to 60 of e^{-0.1x}*x^2

解答

\int_{0}^{60} e^{- 0.1 x} \cdot x^{2} d x

1876.06239 \dots

求解步骤

\int_{0}^{60} e^{- 0.1 x} x^{2} d x

使用分布积分法

= [- 10 e^{- 0.1 x} x^{2} - \int - 20 e^{- 0.1 x} x d x]_{0}^{60}

\int - 20 e^{- 0.1 x} x d x = - 2000 (- 0.1 e^{- 0.1 x} x - e^{- 0.1 x})

= [- 10 e^{- 0.1 x} x^{2} - (- 2000 (- 0.1 e^{- 0.1 x} x - e^{- 0.1 x}))]_{0}^{60}

化简

= [- 10 e^{- 0.1 x} x^{2} + 2000 (- 0.1 e^{- 0.1 x} x - e^{- 0.1 x})]_{0}^{60}

带入上下限计算后得:

1876.06239 \dots

= 1876.06239 \dots

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