Integraal van $$$- \frac{2 x}{\pi} + 1$$$

Bepaal $$$\int \left(- \frac{2 x}{\pi} + 1\right)\, dx$$$.

Integreer termgewijs:

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

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=1$$$:

$$- \int{\frac{2 x}{\pi} d x} + {\color{red}{\int{1 d x}}} = - \int{\frac{2 x}{\pi} d x} + {\color{red}{x}}$$

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

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

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

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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