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Integral de $$$\frac{e^{y}}{x}$$$ con respecto a $$$x$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \frac{e^{y}}{x}\, dx$$$.
Solución
Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=e^{y}$$$ y $$$f{\left(x \right)} = \frac{1}{x}$$$:
$${\color{red}{\int{\frac{e^{y}}{x} d x}}} = {\color{red}{e^{y} \int{\frac{1}{x} d x}}}$$
La integral de $$$\frac{1}{x}$$$ es $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:
$$e^{y} {\color{red}{\int{\frac{1}{x} d x}}} = e^{y} {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$
Por lo tanto,
$$\int{\frac{e^{y}}{x} d x} = e^{y} \ln{\left(\left|{x}\right| \right)}$$
Añade la constante de integración:
$$\int{\frac{e^{y}}{x} d x} = e^{y} \ln{\left(\left|{x}\right| \right)}+C$$
Respuesta
$$$\int \frac{e^{y}}{x}\, dx = e^{y} \ln\left(\left|{x}\right|\right) + C$$$A