Integral of $$$d^{3} x$$$ with respect to $$$x$$$

Find $$$\int d^{3} x\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=d^{3}$$$ and $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{d^{3} x d x}}} = {\color{red}{d^{3} \int{x d x}}}$$

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

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

Therefore,

$$\int{d^{3} x d x} = \frac{d^{3} x^{2}}{2}$$

Add the constant of integration:

$$\int{d^{3} x d x} = \frac{d^{3} x^{2}}{2}+C$$

$$$\int d^{3} x\, dx = \frac{d^{3} x^{2}}{2} + C$$$A