Integraal van $$$2 x^{2} - y^{2}$$$ met betrekking tot $$$x$$$

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

Integreer termgewijs:

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

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=y^{2}$$$:

$$\int{2 x^{2} d x} - {\color{red}{\int{y^{2} d x}}} = \int{2 x^{2} d x} - {\color{red}{x y^{2}}}$$

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

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

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

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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