Integral of $$$6 x^{2}$$$

Find $$$\int 6 x^{2}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=6$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$${\color{red}{\int{6 x^{2} d x}}} = {\color{red}{\left(6 \int{x^{2} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$6 {\color{red}{\int{x^{2} d x}}}=6 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=6 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{6 x^{2} d x} = 2 x^{3}$$

Add the constant of integration:

$$\int{6 x^{2} d x} = 2 x^{3}+C$$

$$$\int 6 x^{2}\, dx = 2 x^{3} + C$$$A