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Does Apostol ever introduce Trig Substitutions ?

  1. Aug 26, 2011 #1
    Does Apostol ever introduce "Trig Substitutions"?

    I took Calculus before, but I am going over Apostol's Calculus Vol. 1 book. In section 5.7 he introduces Integration by Substitution, but never really discusses what was commonly referred to as "Trig Substitutions" in my Calc classes. For instance, #16 in section 5.8 is


    Now I know (from my previous class) that I can solve this by letting [itex]x=tan\theta[/itex], but Apostol never introduced this notion. More importantly, to properly solve it in terms of [itex]x[/itex] I will need to use [itex]arctan[/itex], which isn't introduced until section 6. Just wondering if there's a way to solve it without using trig substitutions.
  2. jcsd
  3. Aug 26, 2011 #2
    Re: Does Apostol ever introduce "Trig Substitutions"?

    Trig substitutions tend to come after u-substitution. Have you looked in other chapters?
  4. Aug 26, 2011 #3
    Re: Does Apostol ever introduce "Trig Substitutions"?

    Yes, section 5.7 is "u-substitution", and section 5.8 contains the exercises which pertain to it. He does several examples of the typical u-substitution methods, but then comes question 16. I skimmed section 6.21, where he introduces the inverse trig functions, but there's mostly integration of the inverse trig functions, not using them as substitution.
  5. Aug 26, 2011 #4
    Re: Does Apostol ever introduce "Trig Substitutions"?

    Got it. He probably wants this solution at this stage:

    Let [itex]u=\frac{x}{\sqrt{x^2+1}}[/itex], then [itex]du=(x^2+1)^\frac{-3}{2}dx[/itex]
    [itex]\int(x^2+1)^\frac{-3}{2}dx=\int du = u + C = \frac{x}{\sqrt{x^2+1}} + C[/itex]

    I suppose that's useful (to be able to recognize that setting u to that value will yield a desirable result), but I hope he does hit trig substitutions at some point.
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