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

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

Simplifique o integrando:

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

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\pi$$$ e $$$f{\left(x \right)} = x \left(2 - x\right)$$$:

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

Expand the expression:

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

Integre termo a termo:

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

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

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

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=2$$$ e $$$f{\left(x \right)} = x$$$:

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

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

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

Portanto,

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

Simplifique:

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

Adicione a constante de integração:

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

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