Integral of $$$m^{2} - n^{2}$$$ with respect to $$$m$$$

Find $$$\int \left(m^{2} - n^{2}\right)\, dm$$$.

Integrate term by term:

$${\color{red}{\int{\left(m^{2} - n^{2}\right)d m}}} = {\color{red}{\left(\int{m^{2} d m} - \int{n^{2} d m}\right)}}$$

Apply the power rule $$$\int m^{n}\, dm = \frac{m^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$- \int{n^{2} d m} + {\color{red}{\int{m^{2} d m}}}=- \int{n^{2} d m} + {\color{red}{\frac{m^{1 + 2}}{1 + 2}}}=- \int{n^{2} d m} + {\color{red}{\left(\frac{m^{3}}{3}\right)}}$$

Apply the constant rule $$$\int c\, dm = c m$$$ with $$$c=n^{2}$$$:

$$\frac{m^{3}}{3} - {\color{red}{\int{n^{2} d m}}} = \frac{m^{3}}{3} - {\color{red}{m n^{2}}}$$

Therefore,

$$\int{\left(m^{2} - n^{2}\right)d m} = \frac{m^{3}}{3} - m n^{2}$$

Simplify:

$$\int{\left(m^{2} - n^{2}\right)d m} = m \left(\frac{m^{2}}{3} - n^{2}\right)$$

Add the constant of integration:

$$\int{\left(m^{2} - n^{2}\right)d m} = m \left(\frac{m^{2}}{3} - n^{2}\right)+C$$

$$$\int \left(m^{2} - n^{2}\right)\, dm = m \left(\frac{m^{2}}{3} - n^{2}\right) + C$$$A