Integral of $$$2 x^{3}$$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = x^{3}$$$:

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

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

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

Therefore,

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

Add the constant of integration:

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

Answer: $$$\int{2 x^{3} d x}=\frac{x^{4}}{2}+C$$$