Integraal van $$$3 x - y$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \left(3 x - y\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(3 x - y\right)d x}}} = {\color{red}{\left(\int{3 x d x} - \int{y d x}\right)}}$$

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

$$\int{3 x d x} - {\color{red}{\int{y d x}}} = \int{3 x d x} - {\color{red}{x y}}$$

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

$$- x y + {\color{red}{\int{3 x d x}}} = - x y + {\color{red}{\left(3 \int{x d x}\right)}}$$

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

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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