Integralen av $$$x^{3} - 2$$$

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

Integrera termvis:

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

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

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

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=3$$$:

$$- 2 x + {\color{red}{\int{x^{3} d x}}}=- 2 x + {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- 2 x + {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Alltså,

$$\int{\left(x^{3} - 2\right)d x} = \frac{x^{4}}{4} - 2 x$$

Förenkla:

$$\int{\left(x^{3} - 2\right)d x} = \frac{x \left(x^{3} - 8\right)}{4}$$

Lägg till integrationskonstanten:

$$\int{\left(x^{3} - 2\right)d x} = \frac{x \left(x^{3} - 8\right)}{4}+C$$

$$$\int \left(x^{3} - 2\right)\, dx = \frac{x \left(x^{3} - 8\right)}{4} + C$$$A