Integral de $$$e^{\frac{2 y}{x}}$$$ con respecto a $$$y$$$

Halla $$$\int e^{\frac{2 y}{x}}\, dy$$$.

Sea $$$u=\frac{2 y}{x}$$$.

Entonces $$$du=\left(\frac{2 y}{x}\right)^{\prime }dy = \frac{2}{x} dy$$$ (los pasos pueden verse »), y obtenemos que $$$dy = \frac{x du}{2}$$$.

Entonces,

$${\color{red}{\int{e^{\frac{2 y}{x}} d y}}} = {\color{red}{\int{\frac{x e^{u}}{2} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{x}{2}$$$ y $$$f{\left(u \right)} = e^{u}$$$:

$${\color{red}{\int{\frac{x e^{u}}{2} d u}}} = {\color{red}{\left(\frac{x \int{e^{u} d u}}{2}\right)}}$$

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

$$\frac{x {\color{red}{\int{e^{u} d u}}}}{2} = \frac{x {\color{red}{e^{u}}}}{2}$$

Recordemos que $$$u=\frac{2 y}{x}$$$:

$$\frac{x e^{{\color{red}{u}}}}{2} = \frac{x e^{{\color{red}{\left(\frac{2 y}{x}\right)}}}}{2}$$

Por lo tanto,

$$\int{e^{\frac{2 y}{x}} d y} = \frac{x e^{\frac{2 y}{x}}}{2}$$

Añade la constante de integración:

$$\int{e^{\frac{2 y}{x}} d y} = \frac{x e^{\frac{2 y}{x}}}{2}+C$$

$$$\int e^{\frac{2 y}{x}}\, dy = \frac{x e^{\frac{2 y}{x}}}{2} + C$$$A