Integral of $$$204 x^{7}$$$

Find $$$\int 204 x^{7}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=204$$$ and $$$f{\left(x \right)} = x^{7}$$$:

$${\color{red}{\int{204 x^{7} d x}}} = {\color{red}{\left(204 \int{x^{7} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=7$$$:

$$204 {\color{red}{\int{x^{7} d x}}}=204 {\color{red}{\frac{x^{1 + 7}}{1 + 7}}}=204 {\color{red}{\left(\frac{x^{8}}{8}\right)}}$$

Therefore,

$$\int{204 x^{7} d x} = \frac{51 x^{8}}{2}$$

Add the constant of integration:

$$\int{204 x^{7} d x} = \frac{51 x^{8}}{2}+C$$

$$$\int 204 x^{7}\, dx = \frac{51 x^{8}}{2} + C$$$A