Integraal van $$$6 x^{5} + 4$$$

Bepaal $$$\int \left(6 x^{5} + 4\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(6 x^{5} + 4\right)d x}}} = {\color{red}{\left(\int{4 d x} + \int{6 x^{5} d x}\right)}}$$

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

$$\int{6 x^{5} d x} + {\color{red}{\int{4 d x}}} = \int{6 x^{5} d x} + {\color{red}{\left(4 x\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=6$$$ en $$$f{\left(x \right)} = x^{5}$$$:

$$4 x + {\color{red}{\int{6 x^{5} d x}}} = 4 x + {\color{red}{\left(6 \int{x^{5} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=5$$$:

$$4 x + 6 {\color{red}{\int{x^{5} d x}}}=4 x + 6 {\color{red}{\frac{x^{1 + 5}}{1 + 5}}}=4 x + 6 {\color{red}{\left(\frac{x^{6}}{6}\right)}}$$

Dus,

$$\int{\left(6 x^{5} + 4\right)d x} = x^{6} + 4 x$$

Vereenvoudig:

$$\int{\left(6 x^{5} + 4\right)d x} = x \left(x^{5} + 4\right)$$

Voeg de integratieconstante toe:

$$\int{\left(6 x^{5} + 4\right)d x} = x \left(x^{5} + 4\right)+C$$

$$$\int \left(6 x^{5} + 4\right)\, dx = x \left(x^{5} + 4\right) + C$$$A