Integralen av $$$x - 5$$$

Bestäm $$$\int \left(x - 5\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(x - 5\right)d x}}} = {\color{red}{\left(- \int{5 d x} + \int{x d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=5$$$:

$$\int{x d x} - {\color{red}{\int{5 d x}}} = \int{x d x} - {\color{red}{\left(5 x\right)}}$$

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

$$- 5 x + {\color{red}{\int{x d x}}}=- 5 x + {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 5 x + {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Alltså,

$$\int{\left(x - 5\right)d x} = \frac{x^{2}}{2} - 5 x$$

Förenkla:

$$\int{\left(x - 5\right)d x} = \frac{x \left(x - 10\right)}{2}$$

Lägg till integrationskonstanten:

$$\int{\left(x - 5\right)d x} = \frac{x \left(x - 10\right)}{2}+C$$

$$$\int \left(x - 5\right)\, dx = \frac{x \left(x - 10\right)}{2} + C$$$A