Integral of $$$\frac{e^{u}}{v}$$$ with respect to $$$u$$$

Find $$$\int \frac{e^{u}}{v}\, du$$$.

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

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

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

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

Therefore,

$$\int{\frac{e^{u}}{v} d u} = \frac{e^{u}}{v}$$

Add the constant of integration:

$$\int{\frac{e^{u}}{v} d u} = \frac{e^{u}}{v}+C$$

$$$\int \frac{e^{u}}{v}\, du = \frac{e^{u}}{v} + C$$$A