Integralen av $$$\frac{8}{t^{2}}$$$

Bestäm $$$\int \frac{8}{t^{2}}\, dt$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ med $$$c=8$$$ och $$$f{\left(t \right)} = \frac{1}{t^{2}}$$$:

$${\color{red}{\int{\frac{8}{t^{2}} d t}}} = {\color{red}{\left(8 \int{\frac{1}{t^{2}} d t}\right)}}$$

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

$$8 {\color{red}{\int{\frac{1}{t^{2}} d t}}}=8 {\color{red}{\int{t^{-2} d t}}}=8 {\color{red}{\frac{t^{-2 + 1}}{-2 + 1}}}=8 {\color{red}{\left(- t^{-1}\right)}}=8 {\color{red}{\left(- \frac{1}{t}\right)}}$$

Alltså,

$$\int{\frac{8}{t^{2}} d t} = - \frac{8}{t}$$

Lägg till integrationskonstanten:

$$\int{\frac{8}{t^{2}} d t} = - \frac{8}{t}+C$$

$$$\int \frac{8}{t^{2}}\, dt = - \frac{8}{t} + C$$$A