Integral of $$$\frac{e^{x}}{x e^{2}}$$$

Find $$$\int \frac{e^{x}}{x e^{2}}\, dx$$$.

The input is rewritten: $$$\int{\frac{e^{x}}{x e^{2}} d x}=\int{\frac{e^{x - 2}}{x} d x}$$$.

Rewrite the integrand:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=e^{-2}$$$ and $$$f{\left(x \right)} = \frac{e^{x}}{x}$$$:

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

This integral (Exponential Integral) does not have a closed form:

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

Therefore,

$$\int{\frac{e^{x - 2}}{x} d x} = \frac{\operatorname{Ei}{\left(x \right)}}{e^{2}}$$

Add the constant of integration:

$$\int{\frac{e^{x - 2}}{x} d x} = \frac{\operatorname{Ei}{\left(x \right)}}{e^{2}}+C$$

$$$\int \frac{e^{x}}{x e^{2}}\, dx = \frac{\operatorname{Ei}{\left(x \right)}}{e^{2}} + C$$$A