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

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

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 $$$f{\left(u \right)} = u$$$:

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

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=1$$$:

$$\frac{{\color{red}{\int{u d u}}}}{v}=\frac{{\color{red}{\frac{u^{1 + 1}}{1 + 1}}}}{v}=\frac{{\color{red}{\left(\frac{u^{2}}{2}\right)}}}{v}$$

Therefore,

$$\int{\frac{u}{v} d u} = \frac{u^{2}}{2 v}$$

Add the constant of integration:

$$\int{\frac{u}{v} d u} = \frac{u^{2}}{2 v}+C$$

$$$\int \frac{u}{v}\, du = \frac{u^{2}}{2 v} + C$$$A