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

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

Integrera termvis:

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

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

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

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

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

Alltså,

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

Förenkla:

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

Lägg till integrationskonstanten:

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

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