Integral of $$$4 x$$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4$$$ and $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{4 x d x}}} = {\color{red}{\left(4 \int{x d x}\right)}}$$

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

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

Therefore,

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

Add the constant of integration:

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

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