Integralen av $$$x^{21} - x$$$
Relaterad kalkylator: Kalkylator för bestämda och oegentliga integraler
Din inmatning
Bestäm $$$\int \left(x^{21} - x\right)\, dx$$$.
Lösning
Integrera termvis:
$${\color{red}{\int{\left(x^{21} - x\right)d x}}} = {\color{red}{\left(- \int{x d x} + \int{x^{21} d x}\right)}}$$
Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=21$$$:
$$- \int{x d x} + {\color{red}{\int{x^{21} d x}}}=- \int{x d x} + {\color{red}{\frac{x^{1 + 21}}{1 + 21}}}=- \int{x d x} + {\color{red}{\left(\frac{x^{22}}{22}\right)}}$$
Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=1$$$:
$$\frac{x^{22}}{22} - {\color{red}{\int{x d x}}}=\frac{x^{22}}{22} - {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{22}}{22} - {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$
Alltså,
$$\int{\left(x^{21} - x\right)d x} = \frac{x^{22}}{22} - \frac{x^{2}}{2}$$
Förenkla:
$$\int{\left(x^{21} - x\right)d x} = \frac{x^{2} \left(x^{20} - 11\right)}{22}$$
Lägg till integrationskonstanten:
$$\int{\left(x^{21} - x\right)d x} = \frac{x^{2} \left(x^{20} - 11\right)}{22}+C$$
Svar
$$$\int \left(x^{21} - x\right)\, dx = \frac{x^{2} \left(x^{20} - 11\right)}{22} + C$$$A