Integralen av $$$x^{17} + x^{7}$$$

Bestäm $$$\int \left(x^{17} + x^{7}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(x^{17} + x^{7}\right)d x}}} = {\color{red}{\left(\int{x^{7} d x} + \int{x^{17} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=7$$$:

$$\int{x^{17} d x} + {\color{red}{\int{x^{7} d x}}}=\int{x^{17} d x} + {\color{red}{\frac{x^{1 + 7}}{1 + 7}}}=\int{x^{17} d x} + {\color{red}{\left(\frac{x^{8}}{8}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=17$$$:

$$\frac{x^{8}}{8} + {\color{red}{\int{x^{17} d x}}}=\frac{x^{8}}{8} + {\color{red}{\frac{x^{1 + 17}}{1 + 17}}}=\frac{x^{8}}{8} + {\color{red}{\left(\frac{x^{18}}{18}\right)}}$$

Alltså,

$$\int{\left(x^{17} + x^{7}\right)d x} = \frac{x^{18}}{18} + \frac{x^{8}}{8}$$

Lägg till integrationskonstanten:

$$\int{\left(x^{17} + x^{7}\right)d x} = \frac{x^{18}}{18} + \frac{x^{8}}{8}+C$$

$$$\int \left(x^{17} + x^{7}\right)\, dx = \left(\frac{x^{18}}{18} + \frac{x^{8}}{8}\right) + C$$$A