Integral de $$$- a^{2} + \frac{1}{s^{2}}$$$ con respecto a $$$a$$$

Halla $$$\int \left(- a^{2} + \frac{1}{s^{2}}\right)\, da$$$.

Integra término a término:

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

Aplica la regla de la constante $$$\int c\, da = a c$$$ con $$$c=\frac{1}{s^{2}}$$$:

$$- \int{a^{2} d a} + {\color{red}{\int{\frac{1}{s^{2}} d a}}} = - \int{a^{2} d a} + {\color{red}{\frac{a}{s^{2}}}}$$

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

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

Por lo tanto,

$$\int{\left(- a^{2} + \frac{1}{s^{2}}\right)d a} = - \frac{a^{3}}{3} + \frac{a}{s^{2}}$$

Añade la constante de integración:

$$\int{\left(- a^{2} + \frac{1}{s^{2}}\right)d a} = - \frac{a^{3}}{3} + \frac{a}{s^{2}}+C$$

$$$\int \left(- a^{2} + \frac{1}{s^{2}}\right)\, da = \left(- \frac{a^{3}}{3} + \frac{a}{s^{2}}\right) + C$$$A