Integraal van $$$5 x^{6} - x^{3} + 2 x^{2} + 21$$$

Bepaal $$$\int \left(5 x^{6} - x^{3} + 2 x^{2} + 21\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(5 x^{6} - x^{3} + 2 x^{2} + 21\right)d x}}} = {\color{red}{\left(\int{21 d x} + \int{2 x^{2} d x} - \int{x^{3} d x} + \int{5 x^{6} d x}\right)}}$$

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=21$$$:

$$\int{2 x^{2} d x} - \int{x^{3} d x} + \int{5 x^{6} d x} + {\color{red}{\int{21 d x}}} = \int{2 x^{2} d x} - \int{x^{3} d x} + \int{5 x^{6} d x} + {\color{red}{\left(21 x\right)}}$$

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

$$21 x + \int{2 x^{2} d x} + \int{5 x^{6} d x} - {\color{red}{\int{x^{3} d x}}}=21 x + \int{2 x^{2} d x} + \int{5 x^{6} d x} - {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=21 x + \int{2 x^{2} d x} + \int{5 x^{6} d x} - {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=2$$$ en $$$f{\left(x \right)} = x^{2}$$$:

$$- \frac{x^{4}}{4} + 21 x + \int{5 x^{6} d x} + {\color{red}{\int{2 x^{2} d x}}} = - \frac{x^{4}}{4} + 21 x + \int{5 x^{6} d x} + {\color{red}{\left(2 \int{x^{2} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=2$$$:

$$- \frac{x^{4}}{4} + 21 x + \int{5 x^{6} d x} + 2 {\color{red}{\int{x^{2} d x}}}=- \frac{x^{4}}{4} + 21 x + \int{5 x^{6} d x} + 2 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \frac{x^{4}}{4} + 21 x + \int{5 x^{6} d x} + 2 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

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

$$- \frac{x^{4}}{4} + \frac{2 x^{3}}{3} + 21 x + {\color{red}{\int{5 x^{6} d x}}} = - \frac{x^{4}}{4} + \frac{2 x^{3}}{3} + 21 x + {\color{red}{\left(5 \int{x^{6} d x}\right)}}$$

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

$$- \frac{x^{4}}{4} + \frac{2 x^{3}}{3} + 21 x + 5 {\color{red}{\int{x^{6} d x}}}=- \frac{x^{4}}{4} + \frac{2 x^{3}}{3} + 21 x + 5 {\color{red}{\frac{x^{1 + 6}}{1 + 6}}}=- \frac{x^{4}}{4} + \frac{2 x^{3}}{3} + 21 x + 5 {\color{red}{\left(\frac{x^{7}}{7}\right)}}$$

Dus,

$$\int{\left(5 x^{6} - x^{3} + 2 x^{2} + 21\right)d x} = \frac{5 x^{7}}{7} - \frac{x^{4}}{4} + \frac{2 x^{3}}{3} + 21 x$$

Vereenvoudig:

$$\int{\left(5 x^{6} - x^{3} + 2 x^{2} + 21\right)d x} = \frac{x \left(60 x^{6} - 21 x^{3} + 56 x^{2} + 1764\right)}{84}$$

Voeg de integratieconstante toe:

$$\int{\left(5 x^{6} - x^{3} + 2 x^{2} + 21\right)d x} = \frac{x \left(60 x^{6} - 21 x^{3} + 56 x^{2} + 1764\right)}{84}+C$$

$$$\int \left(5 x^{6} - x^{3} + 2 x^{2} + 21\right)\, dx = \frac{x \left(60 x^{6} - 21 x^{3} + 56 x^{2} + 1764\right)}{84} + C$$$A