Integralen av $$$\frac{1}{126 t}$$$

Bestäm $$$\int \frac{1}{126 t}\, dt$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ med $$$c=\frac{1}{126}$$$ och $$$f{\left(t \right)} = \frac{1}{t}$$$:

$${\color{red}{\int{\frac{1}{126 t} d t}}} = {\color{red}{\left(\frac{\int{\frac{1}{t} d t}}{126}\right)}}$$

Integralen av $$$\frac{1}{t}$$$ är $$$\int{\frac{1}{t} d t} = \ln{\left(\left|{t}\right| \right)}$$$:

$$\frac{{\color{red}{\int{\frac{1}{t} d t}}}}{126} = \frac{{\color{red}{\ln{\left(\left|{t}\right| \right)}}}}{126}$$

Alltså,

$$\int{\frac{1}{126 t} d t} = \frac{\ln{\left(\left|{t}\right| \right)}}{126}$$

Lägg till integrationskonstanten:

$$\int{\frac{1}{126 t} d t} = \frac{\ln{\left(\left|{t}\right| \right)}}{126}+C$$

$$$\int \frac{1}{126 t}\, dt = \frac{\ln\left(\left|{t}\right|\right)}{126} + C$$$A