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

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

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

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

Entonces,

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

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

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

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

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

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

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

Por lo tanto,

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

Añade la constante de integración:

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

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