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Integration: Trig Substitution

  1. Sep 25, 2005 #1
    When is th ebest time to use it and what are some good rules of thumb for it?
     
  2. jcsd
  3. Sep 25, 2005 #2
    Usually when you an integral in one of these three forms:

    [tex]\int \sqrt{x^2+c^2}dx[/tex]
    [tex]\int \sqrt{x^2-c^2}dx[/tex]
    [tex]\int \sqrt{c^2-x^2}dx[/tex]
     
  4. Sep 25, 2005 #3

    Tom Mattson

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    It's much more general than that, though. The quadratic forms in those integrands need not appear under a radical, and they need not appear in the "numerator line" of an expression (IOW, they can be on the bottom).
     
  5. Sep 26, 2005 #4

    HallsofIvy

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    Generally speaking you should remember that
    cos2x= 1- sin2x
    tan2x= sec2x- 1 and
    sec2x= 1+ tan2x

    Any time you have 1- x2, x2- 1, or 1+ x2 or can reduce to (as, for example 9- x2) you might consider using a trig substitution (unless, of course, something simpler works).
     
  6. Sep 26, 2005 #5

    TD

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    Instead of the tangent-secans relation, you can also use the fact that [itex]\cosh ^2 x - \sinh ^2 x = 1[/itex]. (cp the law with sin and cos, but here with a - instead of a +)
     
  7. Sep 26, 2005 #6
    Yeah. At least for me, that has come up a lot in dealing with arc-length.
     
  8. Sep 26, 2005 #7

    TD

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    Indeed, for example :smile:
     
  9. Sep 26, 2005 #8

    HallsofIvy

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    Yeah, but that wouldn't be a trig substitution, would it!
     
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