Integral de $$$\left(2 x - 2024\right)^{2}$$$

Encontre $$$\int \left(2 x - 2024\right)^{2}\, dx$$$.

Seja $$$u=2 x - 2024$$$.

Então $$$du=\left(2 x - 2024\right)^{\prime }dx = 2 dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{2}$$$.

Portanto,

$${\color{red}{\int{\left(2 x - 2024\right)^{2} d x}}} = {\color{red}{\int{\frac{u^{2}}{2} d u}}}$$

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

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

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

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

Recorde que $$$u=2 x - 2024$$$:

$$\frac{{\color{red}{u}}^{3}}{6} = \frac{{\color{red}{\left(2 x - 2024\right)}}^{3}}{6}$$

Portanto,

$$\int{\left(2 x - 2024\right)^{2} d x} = \frac{\left(2 x - 2024\right)^{3}}{6}$$

Simplifique:

$$\int{\left(2 x - 2024\right)^{2} d x} = \frac{4 \left(x - 1012\right)^{3}}{3}$$

Adicione a constante de integração:

$$\int{\left(2 x - 2024\right)^{2} d x} = \frac{4 \left(x - 1012\right)^{3}}{3}+C$$

$$$\int \left(2 x - 2024\right)^{2}\, dx = \frac{4 \left(x - 1012\right)^{3}}{3} + C$$$A