Integration: Trig Substitution

  • #1

Main Question or Discussion Point

When is th ebest time to use it and what are some good rules of thumb for it?
 

Answers and Replies

  • #2
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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]
 
  • #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).
 
  • #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).
 
  • #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 +)
 
  • #6
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TD said:
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 +)
Yeah. At least for me, that has come up a lot in dealing with arc-length.
 
  • #7
TD
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Indeed, for example :smile:
 
  • #8
HallsofIvy
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TD said:
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 +)
Yeah, but that wouldn't be a trig substitution, would it!
 

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