Integral de $$$e^{1 - x}$$$

Halla $$$\int e^{1 - x}\, dx$$$.

Sea $$$u=1 - x$$$.

Entonces $$$du=\left(1 - x\right)^{\prime }dx = - dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = - du$$$.

Entonces,

$${\color{red}{\int{e^{1 - x} d x}}} = {\color{red}{\int{\left(- e^{u}\right)d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=-1$$$ y $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{\left(- e^{u}\right)d u}}} = {\color{red}{\left(- \int{e^{u} d u}\right)}}$$

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

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

Recordemos que $$$u=1 - x$$$:

$$- e^{{\color{red}{u}}} = - e^{{\color{red}{\left(1 - x\right)}}}$$

Por lo tanto,

$$\int{e^{1 - x} d x} = - e^{1 - x}$$

Añade la constante de integración:

$$\int{e^{1 - x} d x} = - e^{1 - x}+C$$

$$$\int e^{1 - x}\, dx = - e^{1 - x} + C$$$A