Integralen av $$$\frac{\pi}{2}$$$

Bestäm $$$\int \frac{\pi}{2}\, d\pi$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(\pi \right)}\, d\pi = c \int f{\left(\pi \right)}\, d\pi$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(\pi \right)} = \pi$$$:

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

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

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

Alltså,

$$\int{\frac{\pi}{2} d \pi} = \frac{\pi^{2}}{4}$$

Lägg till integrationskonstanten:

$$\int{\frac{\pi}{2} d \pi} = \frac{\pi^{2}}{4}+C$$

$$$\int \frac{\pi}{2}\, d\pi = \frac{\pi^{2}}{4} + C$$$A