Integraal van $$$3 x^{2} - 5 x + 4$$$
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Uw invoer
Bepaal $$$\int \left(3 x^{2} - 5 x + 4\right)\, dx$$$.
Oplossing
Integreer termgewijs:
$${\color{red}{\int{\left(3 x^{2} - 5 x + 4\right)d x}}} = {\color{red}{\left(\int{4 d x} - \int{5 x d x} + \int{3 x^{2} d x}\right)}}$$
Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=4$$$:
$$- \int{5 x d x} + \int{3 x^{2} d x} + {\color{red}{\int{4 d x}}} = - \int{5 x d x} + \int{3 x^{2} d x} + {\color{red}{\left(4 x\right)}}$$
Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=5$$$ en $$$f{\left(x \right)} = x$$$:
$$4 x + \int{3 x^{2} d x} - {\color{red}{\int{5 x d x}}} = 4 x + \int{3 x^{2} d x} - {\color{red}{\left(5 \int{x d x}\right)}}$$
Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=1$$$:
$$4 x + \int{3 x^{2} d x} - 5 {\color{red}{\int{x d x}}}=4 x + \int{3 x^{2} d x} - 5 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=4 x + \int{3 x^{2} d x} - 5 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$
Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=3$$$ en $$$f{\left(x \right)} = x^{2}$$$:
$$- \frac{5 x^{2}}{2} + 4 x + {\color{red}{\int{3 x^{2} d x}}} = - \frac{5 x^{2}}{2} + 4 x + {\color{red}{\left(3 \int{x^{2} d x}\right)}}$$
Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=2$$$:
$$- \frac{5 x^{2}}{2} + 4 x + 3 {\color{red}{\int{x^{2} d x}}}=- \frac{5 x^{2}}{2} + 4 x + 3 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \frac{5 x^{2}}{2} + 4 x + 3 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$
Dus,
$$\int{\left(3 x^{2} - 5 x + 4\right)d x} = x^{3} - \frac{5 x^{2}}{2} + 4 x$$
Vereenvoudig:
$$\int{\left(3 x^{2} - 5 x + 4\right)d x} = \frac{x \left(2 x^{2} - 5 x + 8\right)}{2}$$
Voeg de integratieconstante toe:
$$\int{\left(3 x^{2} - 5 x + 4\right)d x} = \frac{x \left(2 x^{2} - 5 x + 8\right)}{2}+C$$
Antwoord
$$$\int \left(3 x^{2} - 5 x + 4\right)\, dx = \frac{x \left(2 x^{2} - 5 x + 8\right)}{2} + C$$$A