Integralen av $$$- x + 2 \pi$$$

Bestäm $$$\int \left(- x + 2 \pi\right)\, dx$$$.

Integrera termvis:

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

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

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

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

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

Alltså,

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

Förenkla:

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

Lägg till integrationskonstanten:

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

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