Integral of $$$\frac{x^{2}}{a^{2}}$$$ with respect to $$$x$$$

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{a^{2}}$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$${\color{red}{\int{\frac{x^{2}}{a^{2}} d x}}} = {\color{red}{\frac{\int{x^{2} d x}}{a^{2}}}}$$

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

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

Therefore,

$$\int{\frac{x^{2}}{a^{2}} d x} = \frac{x^{3}}{3 a^{2}}$$

Add the constant of integration:

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

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