What is the integral of 1/(x log(x)) dx?

To find the integral of ( \frac{1}{x \log(x)} , dx ), we can utilize a substitution method that simplifies the integral.

Consider the substitution ( u = \log(x) ). This leads to ( du = \frac{1}{x} , dx ), which means ( dx = x , du = e^u , du ). Substituting ( u ) back into the integral, we can rewrite the expression:

[

\int \frac{1}{x \log(x)} , dx = \int \frac{1}{x u} \cdot x , du = \int \frac{1}{u} , du

]

Now, the integral of ( \frac{1}{u} ) is a well-known integral that results in ( \log |u| + C ). Since we set ( u = \log(x) ), substituting back we obtain:

[

\log(|\log(x)|) + C

]

This confirms that the correct answer is indeed the integral of ( \frac{1}{x \log(x)} , dx ).

This method clarifies why the