Integraal van $$$8 e a z - \frac{28 x}{3} - e$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \left(8 e a z - \frac{28 x}{3} - e\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(8 e a z - \frac{28 x}{3} - e\right)d x}}} = {\color{red}{\left(- \int{e d x} - \int{\frac{28 x}{3} d x} + \int{8 e a z d x}\right)}}$$

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

$$- \int{\frac{28 x}{3} d x} + \int{8 e a z d x} - {\color{red}{\int{e d x}}} = - \int{\frac{28 x}{3} d x} + \int{8 e a z d x} - {\color{red}{e x}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{28}{3}$$$ en $$$f{\left(x \right)} = x$$$:

$$- e x + \int{8 e a z d x} - {\color{red}{\int{\frac{28 x}{3} d x}}} = - e x + \int{8 e a z d x} - {\color{red}{\left(\frac{28 \int{x d x}}{3}\right)}}$$

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

$$- e x + \int{8 e a z d x} - \frac{28 {\color{red}{\int{x d x}}}}{3}=- e x + \int{8 e a z d x} - \frac{28 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{3}=- e x + \int{8 e a z d x} - \frac{28 {\color{red}{\left(\frac{x^{2}}{2}\right)}}}{3}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=8 e a z$$$:

$$- \frac{14 x^{2}}{3} - e x + {\color{red}{\int{8 e a z d x}}} = - \frac{14 x^{2}}{3} - e x + {\color{red}{\left(8 e a x z\right)}}$$

Dus,

$$\int{\left(8 e a z - \frac{28 x}{3} - e\right)d x} = 8 e a x z - \frac{14 x^{2}}{3} - e x$$

Vereenvoudig:

$$\int{\left(8 e a z - \frac{28 x}{3} - e\right)d x} = \frac{x \left(24 e a z - 14 x - 3 e\right)}{3}$$

Voeg de integratieconstante toe:

$$\int{\left(8 e a z - \frac{28 x}{3} - e\right)d x} = \frac{x \left(24 e a z - 14 x - 3 e\right)}{3}+C$$

$$$\int \left(8 e a z - \frac{28 x}{3} - e\right)\, dx = \frac{x \left(24 e a z - 14 x - 3 e\right)}{3} + C$$$A