Integral de $$$a^{2} x$$$ con respecto a $$$x$$$

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

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

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

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

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

Por lo tanto,

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

Añade la constante de integración:

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

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