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

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

Integreer termgewijs:

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

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

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

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=g z$$$:

$$- \frac{x^{3}}{3} + {\color{red}{\int{g z d x}}} = - \frac{x^{3}}{3} + {\color{red}{g x z}}$$

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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