Integral de $$$a d e^{\frac{x^{2}}{a^{2}}}$$$ con respecto a $$$x$$$

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

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=a d$$$ y $$$f{\left(x \right)} = e^{\frac{x^{2}}{a^{2}}}$$$:

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

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

Entonces $$$du=\left(\frac{x}{\left|{a}\right|}\right)^{\prime }dx = \frac{dx}{\left|{a}\right|}$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \left|{a}\right| du$$$.

Por lo tanto,

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

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\left|{a}\right|$$$ y $$$f{\left(u \right)} = e^{u^{2}}$$$:

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

Esta integral (Función error imaginaria) no tiene una forma cerrada:

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

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

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

Por lo tanto,

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

Añade la constante de integración:

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

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