Integral of $9 x^{2}$

Find $\int 9 x^{2}\, dx$.

Apply the constant multiple rule $\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$ with $c=9$ and $f{\left(x \right)} = x^{2}$:

$${\color{red}{\int{9 x^{2} d x}}} = {\color{red}{\left(9 \int{x^{2} d x}\right)}}$$

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

$$9 {\color{red}{\int{x^{2} d x}}}=9 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=9 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{9 x^{2} d x} = 3 x^{3}$$

Add the constant of integration:

$$\int{9 x^{2} d x} = 3 x^{3}+C$$

Answer: $\int{9 x^{2} d x}=3 x^{3}+C$