Integraal van $$$b d m o x - 5$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \left(b d m o x - 5\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(b d m o x - 5\right)d x}}} = {\color{red}{\left(- \int{5 d x} + \int{b d m o x d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=5$$$:

$$\int{b d m o x d x} - {\color{red}{\int{5 d x}}} = \int{b d m o x d x} - {\color{red}{\left(5 x\right)}}$$

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

$$- 5 x + {\color{red}{\int{b d m o x d x}}} = - 5 x + {\color{red}{b d m o \int{x d x}}}$$

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

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

Dus,

$$\int{\left(b d m o x - 5\right)d x} = \frac{b d m o x^{2}}{2} - 5 x$$

Vereenvoudig:

$$\int{\left(b d m o x - 5\right)d x} = \frac{x \left(b d m o x - 10\right)}{2}$$

Voeg de integratieconstante toe:

$$\int{\left(b d m o x - 5\right)d x} = \frac{x \left(b d m o x - 10\right)}{2}+C$$

$$$\int \left(b d m o x - 5\right)\, dx = \frac{x \left(b d m o x - 10\right)}{2} + C$$$A