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

Let $$$u=x y$$$.

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

So,

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

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

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

The integral of the exponential function is $$$\int{e^{u} d u} = e^{u}$$$:

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

Recall that $$$u=x y$$$:

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

Therefore,

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

Add the constant of integration:

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

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