Integraal van $$$- 6 x^{6} - 16$$$

Bepaal $$$\int \left(- 6 x^{6} - 16\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(- 6 x^{6} - 16\right)d x}}} = {\color{red}{\left(- \int{16 d x} - \int{6 x^{6} d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=16$$$:

$$- \int{6 x^{6} d x} - {\color{red}{\int{16 d x}}} = - \int{6 x^{6} d x} - {\color{red}{\left(16 x\right)}}$$

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

$$- 16 x - {\color{red}{\int{6 x^{6} d x}}} = - 16 x - {\color{red}{\left(6 \int{x^{6} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=6$$$:

$$- 16 x - 6 {\color{red}{\int{x^{6} d x}}}=- 16 x - 6 {\color{red}{\frac{x^{1 + 6}}{1 + 6}}}=- 16 x - 6 {\color{red}{\left(\frac{x^{7}}{7}\right)}}$$

Dus,

$$\int{\left(- 6 x^{6} - 16\right)d x} = - \frac{6 x^{7}}{7} - 16 x$$

Voeg de integratieconstante toe:

$$\int{\left(- 6 x^{6} - 16\right)d x} = - \frac{6 x^{7}}{7} - 16 x+C$$

$$$\int \left(- 6 x^{6} - 16\right)\, dx = \left(- \frac{6 x^{7}}{7} - 16 x\right) + C$$$A