Integral of $$$5 x^{4}$$$

Find $$$\int 5 x^{4}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=5$$$ and $$$f{\left(x \right)} = x^{4}$$$:

$${\color{red}{\int{5 x^{4} d x}}} = {\color{red}{\left(5 \int{x^{4} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=4$$$:

$$5 {\color{red}{\int{x^{4} d x}}}=5 {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=5 {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

Therefore,

$$\int{5 x^{4} d x} = x^{5}$$

Add the constant of integration:

$$\int{5 x^{4} d x} = x^{5}+C$$

$$$\int 5 x^{4}\, dx = x^{5} + C$$$A