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Integral question

  1. May 23, 2012 #1
    1. The problem statement, all variables and given/known data

    ∫(sec^2 x tanx) dx

    the integral is from 0 to ∏/3

    2. Relevant equations

    I tried using integration by parts


    3. The attempt at a solution

    When I did integration by parts I got to this:

    tan^2 x - ∫(sec^2 x tanx) dx (integral also from 0 to ∏/3)

    which take you back to the question I started with.
     
  2. jcsd
  3. May 23, 2012 #2
    Hint: Are there trigonometric identities that relate [itex]sec^{2}x[/itex] and [itex]tan^{2}x[/itex]?
     
  4. May 23, 2012 #3
    There is a direct u-sub to handle this integral, but the way you have done it does illustrate an important concept.

    Suppose, I = ∫Adx and you perform integration by parts and get I = B - ∫Adx. Well, that second integral is, as you noted, what you started with. Thus, I = B - I and so 2I = B and thus, I=B/2.

    That type of oscillatory behavior is very important for many trigonometric and exponential integrals.
     
  5. May 23, 2012 #4
    Thanks, that was really helpful. I hope I remember and notice it in my test.
     
  6. Jun 5, 2012 #5

    Thanks again! This exact question came up in my math test :)
     
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