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Simplifying an integrand

  1. Apr 4, 2013 #1
    This equality is given in an example, in my textbook:

    [itex]\int\frac{1}{a^2+x^2}dx=\frac{1}{a^2}\int\frac{1}{1+(\frac{x}{a})^2}dx[/itex]

    So, my question is simply... how?
    This is really more of an algebra question, but it'd really help me to see some more detail here.
     
  2. jcsd
  3. Apr 4, 2013 #2
    [itex]\frac{1}{a^2+x^2} = \frac{1}{a^2(1+ \frac{x^2}{a^2})} = \frac{1}{a^2}\frac{1}{1+ (\frac{x}{a})^2}[/itex]
     
  4. Apr 4, 2013 #3
    I see that... sort of. It's a form of factoring, obviously.
    I don't quite get the [itex]\frac{x^2}{a^2}[/itex] bit, though...
    Well, wait: [itex]a^2\times1=a^2[/itex] obviously, and [itex]a^2\times\frac{x^2}{a^2}=x^2[/itex] right? Neat trick, there. Completely obvious too, now that I think about it, but... I guess that I just didn't realize that could be done. Weird.
     
  5. Apr 4, 2013 #4
    Just think of it as factoring out an a2 from the bottom. Here's an easier example:
    [tex]\frac{1}{4+x^{2}}=\frac{1}{(4)1+\frac{x^{2}}{4}}=(\frac{1}{4})\frac{1}{1+{\frac{x^{2}}{4}}}[/tex]
     
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