$$$a d e^{\frac{x^{2}}{a^{2}}}$$$ 关于$$$x$$$的积分

求$$$\int a d e^{\frac{x^{2}}{a^{2}}}\, dx$$$。

对 $$$c=a d$$$ 和 $$$f{\left(x \right)} = e^{\frac{x^{2}}{a^{2}}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{a d e^{\frac{x^{2}}{a^{2}}} d x}}} = {\color{red}{a d \int{e^{\frac{x^{2}}{a^{2}}} d x}}}$$

设$$$u=\frac{x}{\left|{a}\right|}$$$。

则$$$du=\left(\frac{x}{\left|{a}\right|}\right)^{\prime }dx = \frac{dx}{\left|{a}\right|}$$$ (步骤见»)，并有$$$dx = \left|{a}\right| du$$$。

所以，

$$a d {\color{red}{\int{e^{\frac{x^{2}}{a^{2}}} d x}}} = a d {\color{red}{\int{e^{u^{2}} \left|{a}\right| d u}}}$$

对 $$$c=\left|{a}\right|$$$ 和 $$$f{\left(u \right)} = e^{u^{2}}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$a d {\color{red}{\int{e^{u^{2}} \left|{a}\right| d u}}} = a d {\color{red}{\left|{a}\right| \int{e^{u^{2}} d u}}}$$

该积分（虚误差函数）没有闭式表达式：

$$a d \left|{a}\right| {\color{red}{\int{e^{u^{2}} d u}}} = a d \left|{a}\right| {\color{red}{\left(\frac{\sqrt{\pi} \operatorname{erfi}{\left(u \right)}}{2}\right)}}$$

回忆一下 $$$u=\frac{x}{\left|{a}\right|}$$$:

$$\frac{\sqrt{\pi} a d \left|{a}\right| \operatorname{erfi}{\left({\color{red}{u}} \right)}}{2} = \frac{\sqrt{\pi} a d \left|{a}\right| \operatorname{erfi}{\left({\color{red}{\frac{x}{\left|{a}\right|}}} \right)}}{2}$$

因此，

$$\int{a d e^{\frac{x^{2}}{a^{2}}} d x} = \frac{\sqrt{\pi} a d \left|{a}\right| \operatorname{erfi}{\left(\frac{x}{\left|{a}\right|} \right)}}{2}$$

加上积分常数：

$$\int{a d e^{\frac{x^{2}}{a^{2}}} d x} = \frac{\sqrt{\pi} a d \left|{a}\right| \operatorname{erfi}{\left(\frac{x}{\left|{a}\right|} \right)}}{2}+C$$

$$$\int a d e^{\frac{x^{2}}{a^{2}}}\, dx = \frac{\sqrt{\pi} a d \left|{a}\right| \operatorname{erfi}{\left(\frac{x}{\left|{a}\right|} \right)}}{2} + C$$$A