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Integral of $$$\frac{\sqrt{\frac{\ln\left(x\right)}{x}}}{\sqrt{x \ln\left(x\right)}}$$$
Related calculator: Definite and Improper Integral Calculator
Your Input
Find $$$\int \frac{\sqrt{\frac{\ln\left(x\right)}{x}}}{\sqrt{x \ln\left(x\right)}}\, dx$$$.
Solution
The input is rewritten: $$$\int{\frac{\sqrt{\frac{\ln{\left(x \right)}}{x}}}{\sqrt{x \ln{\left(x \right)}}} d x}=\int{\frac{1}{x} d x}$$$.
The integral of $$$\frac{1}{x}$$$ is $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:
$${\color{red}{\int{\frac{1}{x} d x}}} = {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$
Therefore,
$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$
Add the constant of integration:
$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}+C$$
Answer
$$$\int \frac{\sqrt{\frac{\ln\left(x\right)}{x}}}{\sqrt{x \ln\left(x\right)}}\, dx = \ln\left(\left|{x}\right|\right) + C$$$A