Integral of $$$13 \pi h r^{2} v$$$ with respect to $$$v$$$

Find $$$\int 13 \pi h r^{2} v\, dv$$$.

Apply the constant multiple rule $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ with $$$c=13 \pi h r^{2}$$$ and $$$f{\left(v \right)} = v$$$:

$${\color{red}{\int{13 \pi h r^{2} v d v}}} = {\color{red}{\left(13 \pi h r^{2} \int{v d v}\right)}}$$

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

$$13 \pi h r^{2} {\color{red}{\int{v d v}}}=13 \pi h r^{2} {\color{red}{\frac{v^{1 + 1}}{1 + 1}}}=13 \pi h r^{2} {\color{red}{\left(\frac{v^{2}}{2}\right)}}$$

Therefore,

$$\int{13 \pi h r^{2} v d v} = \frac{13 \pi h r^{2} v^{2}}{2}$$

Add the constant of integration:

$$\int{13 \pi h r^{2} v d v} = \frac{13 \pi h r^{2} v^{2}}{2}+C$$

$$$\int 13 \pi h r^{2} v\, dv = \frac{13 \pi h r^{2} v^{2}}{2} + C$$$A