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

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

Sea $$$u=\frac{x}{a}$$$.

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

Por lo tanto,

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

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

$${\color{red}{\int{a e^{u} d u}}} = {\color{red}{a \int{e^{u} d u}}}$$

La integral de la función exponencial es $$$\int{e^{u} d u} = e^{u}$$$:

$$a {\color{red}{\int{e^{u} d u}}} = a {\color{red}{e^{u}}}$$

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

$$a e^{{\color{red}{u}}} = a e^{{\color{red}{\frac{x}{a}}}}$$

Por lo tanto,

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

Añade la constante de integración:

$$\int{e^{\frac{x}{a}} d x} = a e^{\frac{x}{a}}+C$$

$$$\int e^{\frac{x}{a}}\, dx = a e^{\frac{x}{a}} + C$$$A