Integral of $$$g_{3} r^{5}$$$ with respect to $$$g_{3}$$$

Find $$$\int g_{3} r^{5}\, dg_{3}$$$.

Apply the constant multiple rule $$$\int c f{\left(g_{3} \right)}\, dg_{3} = c \int f{\left(g_{3} \right)}\, dg_{3}$$$ with $$$c=r^{5}$$$ and $$$f{\left(g_{3} \right)} = g_{3}$$$:

$${\color{red}{\int{g_{3} r^{5} d g_{3}}}} = {\color{red}{r^{5} \int{g_{3} d g_{3}}}}$$

Apply the power rule $$$\int g_{3}^{n}\, dg_{3} = \frac{g_{3}^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=1$$$:

$$r^{5} {\color{red}{\int{g_{3} d g_{3}}}}=r^{5} {\color{red}{\frac{g_{3}^{1 + 1}}{1 + 1}}}=r^{5} {\color{red}{\left(\frac{g_{3}^{2}}{2}\right)}}$$

Therefore,

$$\int{g_{3} r^{5} d g_{3}} = \frac{g_{3}^{2} r^{5}}{2}$$

Add the constant of integration:

$$\int{g_{3} r^{5} d g_{3}} = \frac{g_{3}^{2} r^{5}}{2}+C$$

$$$\int g_{3} r^{5}\, dg_{3} = \frac{g_{3}^{2} r^{5}}{2} + C$$$A