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Integraal van $$$- 2 x^{3} + x^{2} - 6 x$$$
Gerelateerde rekenmachine: Rekenmachine voor bepaalde en oneigenlijke integralen
Uw invoer
Bepaal $$$\int \left(- 2 x^{3} + x^{2} - 6 x\right)\, dx$$$.
Oplossing
Integreer termgewijs:
$${\color{red}{\int{\left(- 2 x^{3} + x^{2} - 6 x\right)d x}}} = {\color{red}{\left(- \int{6 x d x} + \int{x^{2} d x} - \int{2 x^{3} d x}\right)}}$$
Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=2$$$:
$$- \int{6 x d x} - \int{2 x^{3} d x} + {\color{red}{\int{x^{2} d x}}}=- \int{6 x d x} - \int{2 x^{3} d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \int{6 x d x} - \int{2 x^{3} d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=6$$$ en $$$f{\left(x \right)} = x$$$:
$$\frac{x^{3}}{3} - \int{2 x^{3} d x} - {\color{red}{\int{6 x d x}}} = \frac{x^{3}}{3} - \int{2 x^{3} d x} - {\color{red}{\left(6 \int{x d x}\right)}}$$
Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=1$$$:
$$\frac{x^{3}}{3} - \int{2 x^{3} d x} - 6 {\color{red}{\int{x d x}}}=\frac{x^{3}}{3} - \int{2 x^{3} d x} - 6 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{3}}{3} - \int{2 x^{3} d x} - 6 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$
Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=2$$$ en $$$f{\left(x \right)} = x^{3}$$$:
$$\frac{x^{3}}{3} - 3 x^{2} - {\color{red}{\int{2 x^{3} d x}}} = \frac{x^{3}}{3} - 3 x^{2} - {\color{red}{\left(2 \int{x^{3} d x}\right)}}$$
Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=3$$$:
$$\frac{x^{3}}{3} - 3 x^{2} - 2 {\color{red}{\int{x^{3} d x}}}=\frac{x^{3}}{3} - 3 x^{2} - 2 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=\frac{x^{3}}{3} - 3 x^{2} - 2 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$
Dus,
$$\int{\left(- 2 x^{3} + x^{2} - 6 x\right)d x} = - \frac{x^{4}}{2} + \frac{x^{3}}{3} - 3 x^{2}$$
Vereenvoudig:
$$\int{\left(- 2 x^{3} + x^{2} - 6 x\right)d x} = x^{2} \left(- \frac{x^{2}}{2} + \frac{x}{3} - 3\right)$$
Voeg de integratieconstante toe:
$$\int{\left(- 2 x^{3} + x^{2} - 6 x\right)d x} = x^{2} \left(- \frac{x^{2}}{2} + \frac{x}{3} - 3\right)+C$$
Antwoord
$$$\int \left(- 2 x^{3} + x^{2} - 6 x\right)\, dx = x^{2} \left(- \frac{x^{2}}{2} + \frac{x}{3} - 3\right) + C$$$A