Integral de $$$a d e^{\frac{x^{2}}{a^{2}}}$$$ em relação a $$$x$$$

Encontre $$$\int a d e^{\frac{x^{2}}{a^{2}}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=a d$$$ e $$$f{\left(x \right)} = e^{\frac{x^{2}}{a^{2}}}$$$:

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

Seja $$$u=\frac{x}{\left|{a}\right|}$$$.

Então $$$du=\left(\frac{x}{\left|{a}\right|}\right)^{\prime }dx = \frac{dx}{\left|{a}\right|}$$$ (veja os passos »), e obtemos $$$dx = \left|{a}\right| du$$$.

A integral pode ser reescrita como

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\left|{a}\right|$$$ e $$$f{\left(u \right)} = e^{u^{2}}$$$:

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

Esta integral (Função erro imaginária) não possui forma fechada:

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

Recorde que $$$u=\frac{x}{\left|{a}\right|}$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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