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NDA 2024 Maths · Mathematics · Integrals · medium

∫(dx)/(xln x) is equal to (where C is a constant of integration)

Single correct option

∫dxxln⁡x\displaystyle\int\frac{dx}{x\ln x} is equal to (where CC is a constant of integration)
  1. A. ln⁡∣x∣+C\ln|x|+C
  2. B. 1ln⁡x+C\frac{1}{\ln x}+C
  3. C. ln⁡∣ln⁡x∣+C\ln|\ln x|+C
  4. D. ln⁡∣xln⁡x∣+C\ln|x\ln x|+C
Show answer and solution

Answer: (C) ln⁡∣ln⁡x∣+C\ln|\ln x|+C

Put u=ln⁡xu=\ln x, so du=dxxdu=\frac{dx}{x}. The integral becomes ∫duu=ln⁡∣u∣+C=ln⁡∣ln⁡x∣+C\int\frac{du}{u}=\ln|u|+C=\ln|\ln x|+C. Trap: treating ln⁡x\ln x as a constant gives ln⁡∣x∣\ln|x|; and ln⁡∣xln⁡x∣\ln|x\ln x| differentiates to 1x+1xln⁡x\frac1x+\frac1{x\ln x}, not the integrand.
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