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

解答

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

\frac{e}{2} - \frac{π}{4} erfi (1)

求解步骤

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

化简

x^{2} : xx

= \int_{0}^{1} xx e^{x^{2}} d x

使用分布积分法

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

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

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

带入上下限计算后得:

\frac{e}{2} - \frac{π}{4} erfi (1)

= \frac{e}{2} - \frac{π}{4} erfi (1)

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