Help Understanding Integral with Partial Fractions

  • #1
Hey everyone, I was wondering if someone could help me understand what exactly is happening with a certain integral I am working with, which is as follows:


My steps are as follows (I used partial fractions):

∫(1/(u-1) - 1/(u+1))du = ∫1/(u-1)du - ∫1/(u+1) = ln[(u-1)/(u+1)]

However, here is where my issue arrises; when checking my answer with Mathematica, if I input the very first line above I get:


Could someone help me understand if it is my method that is flawed or maybe the way I am inputting it into the program when I check my answer. Any assistance is greatly appreciated, thanks!
  • #2
ln[(u-1)/(u+1)] = ln[(-1 +u)/(1+u)] = ln[(-1)(1-u)/(1+u)] =ln(-1) + ln[(1-u)/(1+u)] .

Since the integral is indefinite, there must be a constant of integration. The ln(-1) can be incorporated in this constant.

So if you add a constant to your solution, your answer and that given by Matematica will be equivalent.
  • #3
Both answers are correct as they differ by a constant.
  • #4
o ok so the constant can be complex in general then?
  • #5
Yes the constant would be complex. To avoid dealing with this often absolute values are used since otherwise the function would be undefined. In fact different constants are needed in different ranges.
  • #6
thank you very much for the clarification and assistance!

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