Integral de $$$c + f^{2} x^{2}$$$ con respecto a $$$x$$$

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

Integra término a término:

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

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=c$$$:

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

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=f^{2}$$$ y $$$f{\left(x \right)} = x^{2}$$$:

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

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=2$$$:

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

Por lo tanto,

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

Simplificar:

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

Añade la constante de integración:

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

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