Integral of $$$\frac{x^{4}}{7}$$$

Find $$$\int \frac{x^{4}}{7}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{7}$$$ and $$$f{\left(x \right)} = x^{4}$$$:

$${\color{red}{\int{\frac{x^{4}}{7} d x}}} = {\color{red}{\left(\frac{\int{x^{4} d x}}{7}\right)}}$$

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

$$\frac{{\color{red}{\int{x^{4} d x}}}}{7}=\frac{{\color{red}{\frac{x^{1 + 4}}{1 + 4}}}}{7}=\frac{{\color{red}{\left(\frac{x^{5}}{5}\right)}}}{7}$$

Therefore,

$$\int{\frac{x^{4}}{7} d x} = \frac{x^{5}}{35}$$

Add the constant of integration:

$$\int{\frac{x^{4}}{7} d x} = \frac{x^{5}}{35}+C$$

Answer: $$$\int{\frac{x^{4}}{7} d x}=\frac{x^{5}}{35}+C$$$