Integraal van $$$x + y + z$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \left(x + y + z\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(x + y + z\right)d x}}} = {\color{red}{\left(\int{x d x} + \int{y d x} + \int{z d x}\right)}}$$

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

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

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

$$\frac{x^{2}}{2} + \int{z d x} + {\color{red}{\int{y d x}}} = \frac{x^{2}}{2} + \int{z d x} + {\color{red}{x y}}$$

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

$$\frac{x^{2}}{2} + x y + {\color{red}{\int{z d x}}} = \frac{x^{2}}{2} + x y + {\color{red}{x z}}$$

Dus,

$$\int{\left(x + y + z\right)d x} = \frac{x^{2}}{2} + x y + x z$$

Vereenvoudig:

$$\int{\left(x + y + z\right)d x} = \frac{x \left(x + 2 y + 2 z\right)}{2}$$

Voeg de integratieconstante toe:

$$\int{\left(x + y + z\right)d x} = \frac{x \left(x + 2 y + 2 z\right)}{2}+C$$

$$$\int \left(x + y + z\right)\, dx = \frac{x \left(x + 2 y + 2 z\right)}{2} + C$$$A