Integral de $$$\frac{f^{2} x^{2}}{a^{2}}$$$ em relação a $$$x$$$

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

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{f^{2}}{a^{2}}$$$ e $$$f{\left(x \right)} = x^{2}$$$:

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

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=2$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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