Integral de $$$e^{- x y}$$$ con respecto a $$$x$$$

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

Sea $$$u=- x y$$$.

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

Entonces,

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

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

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

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

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

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

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

Por lo tanto,

$$\int{e^{- x y} d x} = - \frac{e^{- x y}}{y}$$

Añade la constante de integración:

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

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