|
|
|
|
|
|
|
|
|
|
|
|
Integraal van $$$- x^{4} + x^{2}$$$
Gerelateerde rekenmachine: Rekenmachine voor bepaalde en oneigenlijke integralen
Uw invoer
Bepaal $$$\int \left(- x^{4} + x^{2}\right)\, dx$$$.
Oplossing
Integreer termgewijs:
$${\color{red}{\int{\left(- x^{4} + x^{2}\right)d x}}} = {\color{red}{\left(\int{x^{2} d x} - \int{x^{4} d x}\right)}}$$
Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=2$$$:
$$- \int{x^{4} d x} + {\color{red}{\int{x^{2} d x}}}=- \int{x^{4} d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \int{x^{4} d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=4$$$:
$$\frac{x^{3}}{3} - {\color{red}{\int{x^{4} d x}}}=\frac{x^{3}}{3} - {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=\frac{x^{3}}{3} - {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$
Dus,
$$\int{\left(- x^{4} + x^{2}\right)d x} = - \frac{x^{5}}{5} + \frac{x^{3}}{3}$$
Voeg de integratieconstante toe:
$$\int{\left(- x^{4} + x^{2}\right)d x} = - \frac{x^{5}}{5} + \frac{x^{3}}{3}+C$$
Antwoord
$$$\int \left(- x^{4} + x^{2}\right)\, dx = \left(- \frac{x^{5}}{5} + \frac{x^{3}}{3}\right) + C$$$A