Integral de $$$\left(1 - x\right)^{2}$$$

Encontre $$$\int \left(1 - x\right)^{2}\, dx$$$.

Seja $$$u=1 - x$$$.

Então $$$du=\left(1 - x\right)^{\prime }dx = - dx$$$ (veja os passos »), e obtemos $$$dx = - du$$$.

Portanto,

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=-1$$$ e $$$f{\left(u \right)} = u^{2}$$$:

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

Aplique a regra da potência $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=2$$$:

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

Recorde que $$$u=1 - x$$$:

$$- \frac{{\color{red}{u}}^{3}}{3} = - \frac{{\color{red}{\left(1 - x\right)}}^{3}}{3}$$

Portanto,

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

Simplifique:

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

Adicione a constante de integração:

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

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