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Techniques of Differentiation - Derivatives of Natural Logarithmic Functions

Techniques of Differentiation - Derivatives of Natural Logarithmic Functions

  1. Express \(\frac{4x}{2x+1}\) in the form \(a + \frac{b}{2x+1}\), where \(a\) and \(b\) are integers.
  2. Differentiate \(2x \ln(2x+1)\) with respect to \(x\).
  3. Using the results in part (a) and part (b), determine \(\displaystyle\displaystyle \displaystyle\int \ln(2x+1) \, \mathrm{d}x\).

Solution:

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Answer:(a) \(a=2\), \(b=-2\) (b) \(2\ln(2x+1)+\dfrac{4x}{2x+1}\) (c) \(\left(x+\dfrac12\right)\ln(2x+1)-x+C\), \(x>-\dfrac12\).

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