Integraal van $$$\pi \left(- x^{2} + 2 x\right)$$$

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

Vereenvoudig de integraand:

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

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\pi$$$ en $$$f{\left(x \right)} = x \left(2 - x\right)$$$:

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

Expand the expression:

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

Integreer termgewijs:

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

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=2$$$:

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

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=2$$$ en $$$f{\left(x \right)} = x$$$:

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

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=1$$$:

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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