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Which integration technique was used here?

  1. Sep 20, 2015 #1
    I've just found the following example on Piskunov:

    [itex]\int \frac{t^2 dt}{(t^2 + 2)^2}\\= \frac{1}{2}\int \frac{t d(t^2 +2)}{(t^2 + 2)^2}\\=-\frac{1}{2}\int t d(\frac{1}{t^2 + 2})\\=-\frac{1}{2}\frac{t}{t^2 + 2}+\int \frac{dt}{t^2 + 2}\\=-\frac{t}{2(t^2 + 2)} + \frac{1}{2 \sqrt{2}}arctan\frac{t}{\sqrt{2}}[/itex]

    What technique is this? Apparently there is a substitution at the beginning, but I can't figure out what happens from second line onwards.
  2. jcsd
  3. Sep 20, 2015 #2


    Staff: Mentor

    The substitution was w = t2 + 2, so dw = 2tdt
    In later steps it appears that they did integration by parts, with u = t and dv = dt/(t2 + 2).
  4. Sep 20, 2015 #3
    Of course! I didn't even try to see an integration by parts there. Thank you!
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