Integral of $$$5 x$$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=5$$$ and $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{5 x d x}}} = {\color{red}{\left(5 \int{x d x}\right)}}$$

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

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

Therefore,

$$\int{5 x d x} = \frac{5 x^{2}}{2}$$

Add the constant of integration:

$$\int{5 x d x} = \frac{5 x^{2}}{2}+C$$

$$$\int 5 x\, dx = \frac{5 x^{2}}{2} + C$$$A