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Integrating ln x by reversing a derivative

Integrating ln x by reversing a derivative

IP 4
7 marks
  1. Differentiate \(x\ln x\) with respect to \(x\), for \(x>0\).[2]
  2. Hence find \(\displaystyle\int\ln x\,\mathrm dx\).[2]
  3. By first differentiating \((2x+1)\ln(2x+1)\), find \(\displaystyle\int\ln(2x+1)\,\mathrm dx\) for \(x>-\dfrac12\).[3]

Solution:

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Answer:(a) \(\ln x+1\) (b) \(x\ln x-x+C\) (c) \(\dfrac12(2x+1)\ln(2x+1)-x+C\)

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