Integraal van $$$41 b d m o x - 3 x - 4$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \left(41 b d m o x - 3 x - 4\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(41 b d m o x - 3 x - 4\right)d x}}} = {\color{red}{\left(- \int{4 d x} - \int{3 x d x} + \int{41 b d m o x d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=4$$$:

$$- \int{3 x d x} + \int{41 b d m o x d x} - {\color{red}{\int{4 d x}}} = - \int{3 x d x} + \int{41 b d m o x d x} - {\color{red}{\left(4 x\right)}}$$

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

$$- 4 x + \int{41 b d m o x d x} - {\color{red}{\int{3 x d x}}} = - 4 x + \int{41 b d m o x d x} - {\color{red}{\left(3 \int{x d x}\right)}}$$

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

$$- 4 x + \int{41 b d m o x d x} - 3 {\color{red}{\int{x d x}}}=- 4 x + \int{41 b d m o x d x} - 3 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 4 x + \int{41 b d m o x d x} - 3 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=41 b d m o$$$ en $$$f{\left(x \right)} = x$$$:

$$- \frac{3 x^{2}}{2} - 4 x + {\color{red}{\int{41 b d m o x d x}}} = - \frac{3 x^{2}}{2} - 4 x + {\color{red}{\left(41 b d m o \int{x d x}\right)}}$$

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

$$41 b d m o {\color{red}{\int{x d x}}} - \frac{3 x^{2}}{2} - 4 x=41 b d m o {\color{red}{\frac{x^{1 + 1}}{1 + 1}}} - \frac{3 x^{2}}{2} - 4 x=41 b d m o {\color{red}{\left(\frac{x^{2}}{2}\right)}} - \frac{3 x^{2}}{2} - 4 x$$

Dus,

$$\int{\left(41 b d m o x - 3 x - 4\right)d x} = \frac{41 b d m o x^{2}}{2} - \frac{3 x^{2}}{2} - 4 x$$

Vereenvoudig:

$$\int{\left(41 b d m o x - 3 x - 4\right)d x} = \frac{x \left(41 b d m o x - 3 x - 8\right)}{2}$$

Voeg de integratieconstante toe:

$$\int{\left(41 b d m o x - 3 x - 4\right)d x} = \frac{x \left(41 b d m o x - 3 x - 8\right)}{2}+C$$

$$$\int \left(41 b d m o x - 3 x - 4\right)\, dx = \frac{x \left(41 b d m o x - 3 x - 8\right)}{2} + C$$$A