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,

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

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 $$$f{\left(u \right)} = e^{u}$$$:

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

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 $$$u=x y$$$:

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

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