Integralen av $$$\frac{2}{35 x}$$$

Bestäm $$$\int \frac{2}{35 x}\, dx$$$.

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{2}{35}$$$ och $$$f{\left(x \right)} = \frac{1}{x}$$$:

$${\color{red}{\int{\frac{2}{35 x} d x}}} = {\color{red}{\left(\frac{2 \int{\frac{1}{x} d x}}{35}\right)}}$$

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

$$\frac{2 {\color{red}{\int{\frac{1}{x} d x}}}}{35} = \frac{2 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}}{35}$$

Alltså,

$$\int{\frac{2}{35 x} d x} = \frac{2 \ln{\left(\left|{x}\right| \right)}}{35}$$

Lägg till integrationskonstanten:

$$\int{\frac{2}{35 x} d x} = \frac{2 \ln{\left(\left|{x}\right| \right)}}{35}+C$$

$$$\int \frac{2}{35 x}\, dx = \frac{2 \ln\left(\left|{x}\right|\right)}{35} + C$$$A