Integral of $$$\frac{e^{y}}{x}$$$ with respect to $$$x$$$

Find $$$\int \frac{e^{y}}{x}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=e^{y}$$$ and $$$f{\left(x \right)} = \frac{1}{x}$$$:

$${\color{red}{\int{\frac{e^{y}}{x} d x}}} = {\color{red}{e^{y} \int{\frac{1}{x} d x}}}$$

The integral of $$$\frac{1}{x}$$$ is $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

$$e^{y} {\color{red}{\int{\frac{1}{x} d x}}} = e^{y} {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Therefore,

$$\int{\frac{e^{y}}{x} d x} = e^{y} \ln{\left(\left|{x}\right| \right)}$$

Add the constant of integration:

$$\int{\frac{e^{y}}{x} d x} = e^{y} \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \frac{e^{y}}{x}\, dx = e^{y} \ln\left(\left|{x}\right|\right) + C$$$A