Integralen av $$$\frac{1}{2 u^{2}}$$$

Bestäm $$$\int \frac{1}{2 u^{2}}\, du$$$.

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

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

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

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

Alltså,

$$\int{\frac{1}{2 u^{2}} d u} = - \frac{1}{2 u}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{2 u^{2}} d u} = - \frac{1}{2 u}+C$$

$$$\int \frac{1}{2 u^{2}}\, du = - \frac{1}{2 u} + C$$$A