Integral de $$$c + f^{2} x^{2}$$$ em relação a $$$x$$$

Encontre $$$\int \left(c + f^{2} x^{2}\right)\, dx$$$.

Integre termo a termo:

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

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=c$$$:

$$\int{f^{2} x^{2} d x} + {\color{red}{\int{c d x}}} = \int{f^{2} x^{2} d x} + {\color{red}{c x}}$$

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

$$c x + {\color{red}{\int{f^{2} x^{2} d x}}} = c x + {\color{red}{f^{2} \int{x^{2} d x}}}$$

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

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

Portanto,

$$\int{\left(c + f^{2} x^{2}\right)d x} = c x + \frac{f^{2} x^{3}}{3}$$

Simplifique:

$$\int{\left(c + f^{2} x^{2}\right)d x} = x \left(c + \frac{f^{2} x^{2}}{3}\right)$$

Adicione a constante de integração:

$$\int{\left(c + f^{2} x^{2}\right)d x} = x \left(c + \frac{f^{2} x^{2}}{3}\right)+C$$

$$$\int \left(c + f^{2} x^{2}\right)\, dx = x \left(c + \frac{f^{2} x^{2}}{3}\right) + C$$$A