Integraal van $$$- a^{2} x^{2} + 1$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \left(- a^{2} x^{2} + 1\right)\, dx$$$.

Integreer termgewijs:

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

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=1$$$:

$$- \int{a^{2} x^{2} d x} + {\color{red}{\int{1 d x}}} = - \int{a^{2} x^{2} d x} + {\color{red}{x}}$$

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

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

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

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

Dus,

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

Voeg de integratieconstante toe:

$$\int{\left(- a^{2} x^{2} + 1\right)d x} = - \frac{a^{2} x^{3}}{3} + x+C$$

$$$\int \left(- a^{2} x^{2} + 1\right)\, dx = \left(- \frac{a^{2} x^{3}}{3} + x\right) + C$$$A