Integralen av $$$1 - x^{3}$$$

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

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

$$- \int{x^{3} d x} + {\color{red}{\int{1 d x}}} = - \int{x^{3} d x} + {\color{red}{x}}$$

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

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

Alltså,

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

Lägg till integrationskonstanten:

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

$$$\int \left(1 - x^{3}\right)\, dx = \left(- \frac{x^{4}}{4} + x\right) + C$$$A