Integral of $$$\frac{d^{3}}{2 \omega}$$$ with respect to $$$d$$$

Find $$$\int \frac{d^{3}}{2 \omega}\, dd$$$.

Apply the constant multiple rule $$$\int c f{\left(d \right)}\, dd = c \int f{\left(d \right)}\, dd$$$ with $$$c=\frac{1}{2 \omega}$$$ and $$$f{\left(d \right)} = d^{3}$$$:

$${\color{red}{\int{\frac{d^{3}}{2 \omega} d d}}} = {\color{red}{\left(\frac{\int{d^{3} d d}}{2 \omega}\right)}}$$

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

$$\frac{{\color{red}{\int{d^{3} d d}}}}{2 \omega}=\frac{{\color{red}{\frac{d^{1 + 3}}{1 + 3}}}}{2 \omega}=\frac{{\color{red}{\left(\frac{d^{4}}{4}\right)}}}{2 \omega}$$

Therefore,

$$\int{\frac{d^{3}}{2 \omega} d d} = \frac{d^{4}}{8 \omega}$$

Add the constant of integration:

$$\int{\frac{d^{3}}{2 \omega} d d} = \frac{d^{4}}{8 \omega}+C$$

$$$\int \frac{d^{3}}{2 \omega}\, dd = \frac{d^{4}}{8 \omega} + C$$$A