Integral de $$$\left(a - x\right)^{2}$$$ con respecto a $$$x$$$

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

Sea $$$u=a - x$$$.

Entonces $$$du=\left(a - x\right)^{\prime }dx = - dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = - du$$$.

La integral puede reescribirse como

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

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=-1$$$ y $$$f{\left(u \right)} = u^{2}$$$:

$${\color{red}{\int{\left(- u^{2}\right)d u}}} = {\color{red}{\left(- \int{u^{2} d u}\right)}}$$

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

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

Recordemos que $$$u=a - x$$$:

$$- \frac{{\color{red}{u}}^{3}}{3} = - \frac{{\color{red}{\left(a - x\right)}}^{3}}{3}$$

Por lo tanto,

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

Simplificar:

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

Añade la constante de integración:

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

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