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Reduction formula, integration problem

  1. Oct 16, 2008 #1
    1. The problem statement, all variables and given/known data

    Part of an example of using reduction formula, I won't post the whole question as I get most of it, just at the very end, magical things happen with the working and things disappear, as they usually do with integration:

    The definite integral goes from 0 to pi/4

    =>[tex]\int[/tex] tan[tex]^{n-2}[/tex] x sec[tex]^{2}[/tex] x dx
    =>tan[tex]^{n-1} x/n -1[/tex]

    3. The attempt at a solution

    My question is, what happened to the sec[tex]^{2}[/tex] x in the integration process?
    I see the tan got integrated, but I can't figure out how sec disappears
     
  2. jcsd
  3. Oct 16, 2008 #2

    Hootenanny

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    Notice that

    [tex]\frac{d}{dx}\tan x = sec^2x[/tex]

    [tex]\Rightarrow dx = \frac{d\left(\tan x\right)}{\sec^2 x}[/tex]
     
  4. Oct 16, 2008 #3
    Ah...I see, so that's how the sec^2 x gets canceled out.

    Thanks hootenanny!
     
  5. Oct 16, 2008 #4

    Hootenanny

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    A pleasure.
     
  6. Oct 16, 2008 #5

    Mark44

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    Since the integral is a definite integral, you should end up with a number, or at least an expression that doesn't involve n. Your final expression can easily be evaluated at both limits of integration.

    As a minor point, you're using an "implies" symbol (==>) incorrectly. The first integral doesn't "imply" the second; it's equal to it.
     
  7. Oct 16, 2008 #6

    Mark44

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    It probably has been asked before, but what's the English equivalent of your signature line? I think I understand a few of the words.

    disce quasi semper victurus vive quasi cras moriturus

    speak? almost always of victory? live almost ?? (you?) die

    Thanks
     
  8. Oct 16, 2008 #7

    Hootenanny

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    I'm impressed! You're quite close to the literal translation, but the meaning is

    "Learn as if you were going to live forever, live as if you were going to die tomorrow".
     
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