Integraal van $$$5 x^{3} - x^{2}$$$

Bepaal $$$\int \left(5 x^{3} - x^{2}\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(5 x^{3} - x^{2}\right)d x}}} = {\color{red}{\left(- \int{x^{2} d x} + \int{5 x^{3} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=2$$$:

$$\int{5 x^{3} d x} - {\color{red}{\int{x^{2} d x}}}=\int{5 x^{3} d x} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\int{5 x^{3} d x} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=5$$$ en $$$f{\left(x \right)} = x^{3}$$$:

$$- \frac{x^{3}}{3} + {\color{red}{\int{5 x^{3} d x}}} = - \frac{x^{3}}{3} + {\color{red}{\left(5 \int{x^{3} d x}\right)}}$$

Pas de machtsregel $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=3$$$:

$$- \frac{x^{3}}{3} + 5 {\color{red}{\int{x^{3} d x}}}=- \frac{x^{3}}{3} + 5 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \frac{x^{3}}{3} + 5 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Dus,

$$\int{\left(5 x^{3} - x^{2}\right)d x} = \frac{5 x^{4}}{4} - \frac{x^{3}}{3}$$

Vereenvoudig:

$$\int{\left(5 x^{3} - x^{2}\right)d x} = \frac{x^{3} \left(15 x - 4\right)}{12}$$

Voeg de integratieconstante toe:

$$\int{\left(5 x^{3} - x^{2}\right)d x} = \frac{x^{3} \left(15 x - 4\right)}{12}+C$$

$$$\int \left(5 x^{3} - x^{2}\right)\, dx = \frac{x^{3} \left(15 x - 4\right)}{12} + C$$$A