|
|
|
|
|
|
|
|
|
|
|
|
Integralen av $$$- x^{3} + x^{2}$$$
Relaterad kalkylator: Kalkylator för bestämda och oegentliga integraler
Din inmatning
Bestäm $$$\int \left(- x^{3} + x^{2}\right)\, dx$$$.
Lösning
Integrera termvis:
$${\color{red}{\int{\left(- x^{3} + x^{2}\right)d x}}} = {\color{red}{\left(\int{x^{2} d x} - \int{x^{3} d x}\right)}}$$
Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:
$$- \int{x^{3} d x} + {\color{red}{\int{x^{2} d x}}}=- \int{x^{3} d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \int{x^{3} d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=3$$$:
$$\frac{x^{3}}{3} - {\color{red}{\int{x^{3} d x}}}=\frac{x^{3}}{3} - {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=\frac{x^{3}}{3} - {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$
Alltså,
$$\int{\left(- x^{3} + x^{2}\right)d x} = - \frac{x^{4}}{4} + \frac{x^{3}}{3}$$
Förenkla:
$$\int{\left(- x^{3} + x^{2}\right)d x} = \frac{x^{3} \left(4 - 3 x\right)}{12}$$
Lägg till integrationskonstanten:
$$\int{\left(- x^{3} + x^{2}\right)d x} = \frac{x^{3} \left(4 - 3 x\right)}{12}+C$$
Svar
$$$\int \left(- x^{3} + x^{2}\right)\, dx = \frac{x^{3} \left(4 - 3 x\right)}{12} + C$$$A