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Reduction formula, integration problem

  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. Hootenanny

    Hootenanny 9,681
    Staff Emeritus
    Science Advisor
    Gold Member

    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. Ah...I see, so that's how the sec^2 x gets canceled out.

    Thanks hootenanny!
     
  5. Hootenanny

    Hootenanny 9,681
    Staff Emeritus
    Science Advisor
    Gold Member

    A pleasure.
     
  6. Mark44

    Staff: Mentor

    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. Mark44

    Staff: Mentor

    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. Hootenanny

    Hootenanny 9,681
    Staff Emeritus
    Science Advisor
    Gold Member

    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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