Integraal van $$$- \frac{x^{2} \left(3 - \frac{1}{x^{2}}\right)}{3}$$$

Bepaal $$$\int \left(- \frac{x^{2} \left(3 - \frac{1}{x^{2}}\right)}{3}\right)\, dx$$$.

De invoer is herschreven: $$$\int{\left(- \frac{x^{2} \left(3 - \frac{1}{x^{2}}\right)}{3}\right)d x}=\int{x^{2} \left(-1 + \frac{1}{3 x^{2}}\right) d x}$$$.

Vereenvoudig de integraand:

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

Integreer termgewijs:

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

Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=\frac{1}{3}$$$:

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

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

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

Dus,

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

Vereenvoudig:

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

Voeg de integratieconstante toe:

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

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