Integraal van $$$- 10 x^{53} + \frac{x^{6}}{2}$$$

Bepaal $$$\int \left(- 10 x^{53} + \frac{x^{6}}{2}\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(- 10 x^{53} + \frac{x^{6}}{2}\right)d x}}} = {\color{red}{\left(\int{\frac{x^{6}}{2} d x} - \int{10 x^{53} d x}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(x \right)} = x^{6}$$$:

$$- \int{10 x^{53} d x} + {\color{red}{\int{\frac{x^{6}}{2} d x}}} = - \int{10 x^{53} d x} + {\color{red}{\left(\frac{\int{x^{6} d x}}{2}\right)}}$$

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

$$- \int{10 x^{53} d x} + \frac{{\color{red}{\int{x^{6} d x}}}}{2}=- \int{10 x^{53} d x} + \frac{{\color{red}{\frac{x^{1 + 6}}{1 + 6}}}}{2}=- \int{10 x^{53} d x} + \frac{{\color{red}{\left(\frac{x^{7}}{7}\right)}}}{2}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=10$$$ en $$$f{\left(x \right)} = x^{53}$$$:

$$\frac{x^{7}}{14} - {\color{red}{\int{10 x^{53} d x}}} = \frac{x^{7}}{14} - {\color{red}{\left(10 \int{x^{53} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=53$$$:

$$\frac{x^{7}}{14} - 10 {\color{red}{\int{x^{53} d x}}}=\frac{x^{7}}{14} - 10 {\color{red}{\frac{x^{1 + 53}}{1 + 53}}}=\frac{x^{7}}{14} - 10 {\color{red}{\left(\frac{x^{54}}{54}\right)}}$$

Dus,

$$\int{\left(- 10 x^{53} + \frac{x^{6}}{2}\right)d x} = - \frac{5 x^{54}}{27} + \frac{x^{7}}{14}$$

Vereenvoudig:

$$\int{\left(- 10 x^{53} + \frac{x^{6}}{2}\right)d x} = \frac{x^{7} \left(27 - 70 x^{47}\right)}{378}$$

Voeg de integratieconstante toe:

$$\int{\left(- 10 x^{53} + \frac{x^{6}}{2}\right)d x} = \frac{x^{7} \left(27 - 70 x^{47}\right)}{378}+C$$

$$$\int \left(- 10 x^{53} + \frac{x^{6}}{2}\right)\, dx = \frac{x^{7} \left(27 - 70 x^{47}\right)}{378} + C$$$A