Integraal van $$$i a n t x^{3} - 7$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \left(i a n t x^{3} - 7\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(i a n t x^{3} - 7\right)d x}}} = {\color{red}{\left(- \int{7 d x} + \int{i a n t x^{3} d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=7$$$:

$$\int{i a n t x^{3} d x} - {\color{red}{\int{7 d x}}} = \int{i a n t x^{3} d x} - {\color{red}{\left(7 x\right)}}$$

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

$$- 7 x + {\color{red}{\int{i a n t x^{3} d x}}} = - 7 x + {\color{red}{i a n t \int{x^{3} d x}}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=3$$$:

$$i a n t {\color{red}{\int{x^{3} d x}}} - 7 x=i a n t {\color{red}{\frac{x^{1 + 3}}{1 + 3}}} - 7 x=i a n t {\color{red}{\left(\frac{x^{4}}{4}\right)}} - 7 x$$

Dus,

$$\int{\left(i a n t x^{3} - 7\right)d x} = \frac{i a n t x^{4}}{4} - 7 x$$

Vereenvoudig:

$$\int{\left(i a n t x^{3} - 7\right)d x} = \frac{x \left(i a n t x^{3} - 28\right)}{4}$$

Voeg de integratieconstante toe:

$$\int{\left(i a n t x^{3} - 7\right)d x} = \frac{x \left(i a n t x^{3} - 28\right)}{4}+C$$

$$$\int \left(i a n t x^{3} - 7\right)\, dx = \frac{x \left(i a n t x^{3} - 28\right)}{4} + C$$$A