Integral of $$$12 x^{5}$$$

Find $$$\int 12 x^{5}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=12$$$ and $$$f{\left(x \right)} = x^{5}$$$:

$${\color{red}{\int{12 x^{5} d x}}} = {\color{red}{\left(12 \int{x^{5} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=5$$$:

$$12 {\color{red}{\int{x^{5} d x}}}=12 {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=12 {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

Therefore,

$$\int{12 x^{5} d x} = 2 x^{6}$$

Add the constant of integration:

$$\int{12 x^{5} d x} = 2 x^{6}+C$$

$$$\int 12 x^{5}\, dx = 2 x^{6} + C$$$A