Integralen av $$$x^{2} - 9$$$

Bestäm $$$\int \left(x^{2} - 9\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(x^{2} - 9\right)d x}}} = {\color{red}{\left(- \int{9 d x} + \int{x^{2} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=9$$$:

$$\int{x^{2} d x} - {\color{red}{\int{9 d x}}} = \int{x^{2} d x} - {\color{red}{\left(9 x\right)}}$$

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

$$- 9 x + {\color{red}{\int{x^{2} d x}}}=- 9 x + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- 9 x + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Alltså,

$$\int{\left(x^{2} - 9\right)d x} = \frac{x^{3}}{3} - 9 x$$

Förenkla:

$$\int{\left(x^{2} - 9\right)d x} = \frac{x \left(x^{2} - 27\right)}{3}$$

Lägg till integrationskonstanten:

$$\int{\left(x^{2} - 9\right)d x} = \frac{x \left(x^{2} - 27\right)}{3}+C$$

$$$\int \left(x^{2} - 9\right)\, dx = \frac{x \left(x^{2} - 27\right)}{3} + C$$$A