Integralen av $$$\frac{\sqrt{\frac{\ln\left(x\right)}{x}}}{\sqrt{x \ln\left(x\right)}}$$$

Bestäm $$$\int \frac{\sqrt{\frac{\ln\left(x\right)}{x}}}{\sqrt{x \ln\left(x\right)}}\, dx$$$.

Inmatningen skrivs om: $$$\int{\frac{\sqrt{\frac{\ln{\left(x \right)}}{x}}}{\sqrt{x \ln{\left(x \right)}}} d x}=\int{\frac{1}{x} d x}$$$.

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

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

Alltså,

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

Lägg till integrationskonstanten:

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

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