Integralen av $$$2 - x^{2}$$$

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

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=2$$$:

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

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:

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

Alltså,

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

Förenkla:

$$\int{\left(2 - x^{2}\right)d x} = \frac{x \left(6 - x^{2}\right)}{3}$$

Lägg till integrationskonstanten:

$$\int{\left(2 - x^{2}\right)d x} = \frac{x \left(6 - x^{2}\right)}{3}+C$$

$$$\int \left(2 - x^{2}\right)\, dx = \frac{x \left(6 - x^{2}\right)}{3} + C$$$A