Integraal van $$$\frac{24}{5} - \frac{6 x}{5}$$$

Bepaal $$$\int \left(\frac{24}{5} - \frac{6 x}{5}\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(\frac{24}{5} - \frac{6 x}{5}\right)d x}}} = {\color{red}{\left(\int{\frac{24}{5} d x} - \int{\frac{6 x}{5} d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=\frac{24}{5}$$$:

$$- \int{\frac{6 x}{5} d x} + {\color{red}{\int{\frac{24}{5} d x}}} = - \int{\frac{6 x}{5} d x} + {\color{red}{\left(\frac{24 x}{5}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{6}{5}$$$ en $$$f{\left(x \right)} = x$$$:

$$\frac{24 x}{5} - {\color{red}{\int{\frac{6 x}{5} d x}}} = \frac{24 x}{5} - {\color{red}{\left(\frac{6 \int{x d x}}{5}\right)}}$$

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

$$\frac{24 x}{5} - \frac{6 {\color{red}{\int{x d x}}}}{5}=\frac{24 x}{5} - \frac{6 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{5}=\frac{24 x}{5} - \frac{6 {\color{red}{\left(\frac{x^{2}}{2}\right)}}}{5}$$

Dus,

$$\int{\left(\frac{24}{5} - \frac{6 x}{5}\right)d x} = - \frac{3 x^{2}}{5} + \frac{24 x}{5}$$

Vereenvoudig:

$$\int{\left(\frac{24}{5} - \frac{6 x}{5}\right)d x} = \frac{3 x \left(8 - x\right)}{5}$$

Voeg de integratieconstante toe:

$$\int{\left(\frac{24}{5} - \frac{6 x}{5}\right)d x} = \frac{3 x \left(8 - x\right)}{5}+C$$

$$$\int \left(\frac{24}{5} - \frac{6 x}{5}\right)\, dx = \frac{3 x \left(8 - x\right)}{5} + C$$$A