Question

1. $\int\frac{dx}{x\ln x}$

Explanation:

Step1: Let \( u = \ln x \)

Differentiate \( u \) with respect to \( x \): \( du=\frac{1}{x}dx \)

Step2: Substitute into the integral

The integral \( \int\frac{dx}{x\ln x} \) becomes \( \int\frac{du}{u} \)

Step3: Integrate \( \int\frac{du}{u} \)

Using the formula \( \int\frac{1}{u}du=\ln|u| + C \), we get \( \ln|u|+C \)

Step4: Substitute back \( u = \ln x \)

We have \( \ln|\ln x|+C \)

Answer:

\(\ln|\ln x| + C\)