Integraal van $$$2 x^{3} - 35 x$$$

Bepaal $$$\int \left(2 x^{3} - 35 x\right)\, dx$$$.

Integreer termgewijs:

$${\color{red}{\int{\left(2 x^{3} - 35 x\right)d x}}} = {\color{red}{\left(- \int{35 x d x} + \int{2 x^{3} d x}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=35$$$ en $$$f{\left(x \right)} = x$$$:

$$\int{2 x^{3} d x} - {\color{red}{\int{35 x d x}}} = \int{2 x^{3} d x} - {\color{red}{\left(35 \int{x d x}\right)}}$$

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

$$\int{2 x^{3} d x} - 35 {\color{red}{\int{x d x}}}=\int{2 x^{3} d x} - 35 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\int{2 x^{3} d x} - 35 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=2$$$ en $$$f{\left(x \right)} = x^{3}$$$:

$$- \frac{35 x^{2}}{2} + {\color{red}{\int{2 x^{3} d x}}} = - \frac{35 x^{2}}{2} + {\color{red}{\left(2 \int{x^{3} d x}\right)}}$$

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

$$- \frac{35 x^{2}}{2} + 2 {\color{red}{\int{x^{3} d x}}}=- \frac{35 x^{2}}{2} + 2 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- \frac{35 x^{2}}{2} + 2 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Dus,

$$\int{\left(2 x^{3} - 35 x\right)d x} = \frac{x^{4}}{2} - \frac{35 x^{2}}{2}$$

Vereenvoudig:

$$\int{\left(2 x^{3} - 35 x\right)d x} = \frac{x^{2} \left(x^{2} - 35\right)}{2}$$

Voeg de integratieconstante toe:

$$\int{\left(2 x^{3} - 35 x\right)d x} = \frac{x^{2} \left(x^{2} - 35\right)}{2}+C$$

$$$\int \left(2 x^{3} - 35 x\right)\, dx = \frac{x^{2} \left(x^{2} - 35\right)}{2} + C$$$A