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

Halla $$$\int \left(a^{2} t - x^{2}\right)\, dx$$$.

Integra término a término:

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

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

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

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=a^{2} t$$$:

$$- \frac{x^{3}}{3} + {\color{red}{\int{a^{2} t d x}}} = - \frac{x^{3}}{3} + {\color{red}{a^{2} t x}}$$

Por lo tanto,

$$\int{\left(a^{2} t - x^{2}\right)d x} = a^{2} t x - \frac{x^{3}}{3}$$

Simplificar:

$$\int{\left(a^{2} t - x^{2}\right)d x} = x \left(a^{2} t - \frac{x^{2}}{3}\right)$$

Añade la constante de integración:

$$\int{\left(a^{2} t - x^{2}\right)d x} = x \left(a^{2} t - \frac{x^{2}}{3}\right)+C$$

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