integral from 0 to 0.5 of x^2e^{-x^2}

解答

\int_{0}^{0.5} x^{2} e^{- x^{2}} d x

0.03594 \dots

求解步骤

\int_{0}^{0.5} x^{2} e^{- x^{2}} d x

化简

x^{2} : xx

= \int_{0}^{0.5} xx e^{- x^{2}} d x

使用分布积分法

= [\frac{1}{2} (- e^{- x^{2}} x - 2 \cdot \int - \frac{1}{2} e^{- x^{2}} d x)]_{0}^{0.5}

\int - \frac{1}{2} e^{- x^{2}} d x = - \frac{π}{4} erf (x)

= [\frac{1}{2} (- e^{- x^{2}} x - 2 (- \frac{π}{4} erf (x)))]_{0}^{0.5}

化简

[\frac{1}{2} (- e^{- x^{2}} x - 2 (- \frac{π}{4} erf (x)))]_{0}^{0.5} : [0.5 (- e^{- x^{2}} x + 0.88622 \dots erf (x))]_{0}^{0.5}

= [0.5 (- e^{- x^{2}} x + 0.88622 \dots erf (x))]_{0}^{0.5}

带入上下限计算后得:

0.03594 \dots

= 0.03594 \dots

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