integral from x/2 to 1/2 of x^2e^{y^2}

解答

\int_{\frac{x}{2}}^{\frac{1}{2}} x^{2} e^{y^{2}} d y

x^{2} (\frac{π}{2} erfi (\frac{1}{2}) - \frac{π}{2} erfi (\frac{x}{2}))

求解步骤

\int_{\frac{x}{2}}^{\frac{1}{2}} x^{2} e^{y^{2}} d y

提出常数:

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

= x^{2} \cdot \int_{\frac{x}{2}}^{\frac{1}{2}} e^{y^{2}} d y

这是非初等函数积分:

\int e^{y^{2}} d y = \frac{π}{2} erfi (y)

= x^{2} [\frac{π}{2} erfi (y)]_{\frac{x}{2}}^{\frac{1}{2}}

带入上下限计算后得:

\frac{π}{2} erfi (\frac{1}{2}) - \frac{π}{2} erfi (\frac{x}{2})

= x^{2} (\frac{π}{2} erfi (\frac{1}{2}) - \frac{π}{2} erfi (\frac{x}{2}))

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