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

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

For the integral $$$\int{e^{\frac{y}{x}} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=e^{\frac{y}{x}}$$$ and $$$\operatorname{dv}=dx$$$.

Then $$$\operatorname{du}=\left(e^{\frac{y}{x}}\right)^{\prime }dx=- \frac{y e^{\frac{y}{x}}}{x^{2}} dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{1 d x}=x$$$ (steps can be seen »).

Thus,

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

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

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

Let $$$u=\frac{y}{x}$$$.

Then $$$du=\left(\frac{y}{x}\right)^{\prime }dx = - \frac{y}{x^{2}} dx$$$ (steps can be seen »), and we have that $$$\frac{dx}{x^{2}} = - \frac{du}{y}$$$.

The integral becomes

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

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

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

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

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

Recall that $$$u=\frac{y}{x}$$$:

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

Therefore,

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

Add the constant of integration:

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

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