Integral of $$$u v$$$ with respect to $$$u$$$

Find $$$\int u v\, du$$$.

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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