Question Detail
Question
Integrate ∫ dx / (x - x ln x).
Correct Answer
Option D is the correct answer.
Detailed Explanation
First, factor out x from the denominator: ∫ dx / (x(1 - ln x)).Now, use substitution. Let u = 1 - ln x.Then, du = d/dx (1 - ln x) dx = - (1/x) dx.This means dx/x = -du.Substitute these into the integral:∫ (-du) / u = -∫ (1/u) du = -ln|u| + C.Substitute back u = 1 - ln x:-ln|1 - ln x| + C.Assuming the domain where 1 - ln x > 0, the absolute value can be removed: -ln(1 - ln x) + C.
Hint
Look for a substitution that simplifies the denominator. Factoring out x first can reveal a suitable substitution involving ln x.