|
|
|
|
|
|
|
|
|
|
|
|
Integralen av $$$- x^{22} + x^{7}$$$
Relaterad kalkylator: Kalkylator för bestämda och oegentliga integraler
Din inmatning
Bestäm $$$\int \left(- x^{22} + x^{7}\right)\, dx$$$.
Lösning
Integrera termvis:
$${\color{red}{\int{\left(- x^{22} + x^{7}\right)d x}}} = {\color{red}{\left(\int{x^{7} d x} - \int{x^{22} d x}\right)}}$$
Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=7$$$:
$$- \int{x^{22} d x} + {\color{red}{\int{x^{7} d x}}}=- \int{x^{22} d x} + {\color{red}{\frac{x^{1 + 7}}{1 + 7}}}=- \int{x^{22} d x} + {\color{red}{\left(\frac{x^{8}}{8}\right)}}$$
Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=22$$$:
$$\frac{x^{8}}{8} - {\color{red}{\int{x^{22} d x}}}=\frac{x^{8}}{8} - {\color{red}{\frac{x^{1 + 22}}{1 + 22}}}=\frac{x^{8}}{8} - {\color{red}{\left(\frac{x^{23}}{23}\right)}}$$
Alltså,
$$\int{\left(- x^{22} + x^{7}\right)d x} = - \frac{x^{23}}{23} + \frac{x^{8}}{8}$$
Lägg till integrationskonstanten:
$$\int{\left(- x^{22} + x^{7}\right)d x} = - \frac{x^{23}}{23} + \frac{x^{8}}{8}+C$$
Svar
$$$\int \left(- x^{22} + x^{7}\right)\, dx = \left(- \frac{x^{23}}{23} + \frac{x^{8}}{8}\right) + C$$$A