Integral of $$$\frac{m}{s}$$$ with respect to $$$m$$$

Find $$$\int \frac{m}{s}\, dm$$$.

Apply the constant multiple rule $$$\int c f{\left(m \right)}\, dm = c \int f{\left(m \right)}\, dm$$$ with $$$c=\frac{1}{s}$$$ and $$$f{\left(m \right)} = m$$$:

$${\color{red}{\int{\frac{m}{s} d m}}} = {\color{red}{\frac{\int{m d m}}{s}}}$$

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

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

Therefore,

$$\int{\frac{m}{s} d m} = \frac{m^{2}}{2 s}$$

Add the constant of integration:

$$\int{\frac{m}{s} d m} = \frac{m^{2}}{2 s}+C$$

$$$\int \frac{m}{s}\, dm = \frac{m^{2}}{2 s} + C$$$A