$$$3^{x^{2}}$$$ 的积分

求$$$\int 3^{x^{2}}\, dx$$$。

换底:

$${\color{red}{\int{3^{x^{2}} d x}}} = {\color{red}{\int{e^{x^{2} \ln{\left(3 \right)}} d x}}}$$

设$$$u=x \sqrt{\ln{\left(3 \right)}}$$$。

则$$$du=\left(x \sqrt{\ln{\left(3 \right)}}\right)^{\prime }dx = \sqrt{\ln{\left(3 \right)}} dx$$$ (步骤见»)，并有$$$dx = \frac{du}{\sqrt{\ln{\left(3 \right)}}}$$$。

因此，

$${\color{red}{\int{e^{x^{2} \ln{\left(3 \right)}} d x}}} = {\color{red}{\int{\frac{e^{u^{2}}}{\sqrt{\ln{\left(3 \right)}}} d u}}}$$

对 $$$c=\frac{1}{\sqrt{\ln{\left(3 \right)}}}$$$ 和 $$$f{\left(u \right)} = e^{u^{2}}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$${\color{red}{\int{\frac{e^{u^{2}}}{\sqrt{\ln{\left(3 \right)}}} d u}}} = {\color{red}{\frac{\int{e^{u^{2}} d u}}{\sqrt{\ln{\left(3 \right)}}}}}$$

该积分（虚误差函数）没有闭式表达式：

$$\frac{{\color{red}{\int{e^{u^{2}} d u}}}}{\sqrt{\ln{\left(3 \right)}}} = \frac{{\color{red}{\left(\frac{\sqrt{\pi} \operatorname{erfi}{\left(u \right)}}{2}\right)}}}{\sqrt{\ln{\left(3 \right)}}}$$

回忆一下 $$$u=x \sqrt{\ln{\left(3 \right)}}$$$:

$$\frac{\sqrt{\pi} \operatorname{erfi}{\left({\color{red}{u}} \right)}}{2 \sqrt{\ln{\left(3 \right)}}} = \frac{\sqrt{\pi} \operatorname{erfi}{\left({\color{red}{x \sqrt{\ln{\left(3 \right)}}}} \right)}}{2 \sqrt{\ln{\left(3 \right)}}}$$

因此，

$$\int{3^{x^{2}} d x} = \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \sqrt{\ln{\left(3 \right)}} \right)}}{2 \sqrt{\ln{\left(3 \right)}}}$$

加上积分常数：

$$\int{3^{x^{2}} d x} = \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \sqrt{\ln{\left(3 \right)}} \right)}}{2 \sqrt{\ln{\left(3 \right)}}}+C$$

$$$\int 3^{x^{2}}\, dx = \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \sqrt{\ln\left(3\right)} \right)}}{2 \sqrt{\ln\left(3\right)}} + C$$$A