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Change of variables in an integral

  1. Jul 10, 2015 #1
    This is not really a homework or anything, just found myself hitting a wall when doodling around.
    If I have an integral like

    ##\int_{-1}^{0} x(x^2-1) dx##

    and I introduce a new variable:
    ##u = x^2##
    How do I calculate the limits of the new integral? In this case the upper limit is obviously 0, but what about the lower limit?
     
  2. jcsd
  3. Jul 10, 2015 #2
    For the lower limit you plug in -1 in that u=x^2 and so you have u=1... same thing for the upper limit. Of course you don't necessarily have to change the variable for this integral......
     
  4. Jul 10, 2015 #3

    HallsofIvy

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    When x= -1, u= (-1)2= 1, of course. If it bothers you that the lower limit of integration is larger than the upper limit just use the fact that [itex]\int_a^b f(x)dx= -\int_b^a f(x)dx[/itex].
     
  5. Jul 10, 2015 #4
    Argh, that was a brain fart.. But what about a case where
    ##\int_{-1}^{0} x dx##
    and I make a change of variables like:
    ##u^2 = x##
    Does the lower limit become a complex number?
     
  6. Jul 11, 2015 #5

    HallsofIvy

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    Yes, and, in fact, you would have to consider the case of u= i and u= -i separately. Further you would have to consider that in the complex numbers there are an infinite number of paths between i and 0 or between -i and 0! Not every substitution is a good idea.
     
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