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Reduction Formulae Question : In= ∫x(cos^n(x))

  1. Apr 12, 2017 #1
    1. The problem statement, all variables and given/known data
    Let In = ∫x(cos^n(x)) with limits between x=π/2, x=0 for n≥0
    i) Show that nIn=(n-1)In-2 -n^-1 for n≥2
    ii) Find the exact value of I3

    2. Relevant equations
    ∫u'v = uv-∫uv' is what I use for these questions

    3. The attempt at a solution
    Rewritten as ∫ xcos^n-1(x) cosx
    u'=cosx v= xcos^n-1x
    u= sinx v'= cos^n-1 -x(n-1)sinxcos^n-2x
    But I can't seem to write it in the form it asks for.
     
  2. jcsd
  3. Apr 12, 2017 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Your questions are indecipherable. Strange notation and no parentheses.
     
  4. Apr 12, 2017 #3

    Mark44

    Staff: Mentor

    Part of the problem is that if you type i in brackets, the browser things you mean the change the font type to italics. Besides that, it's hard to tell exactly what the problem is you're trying to solve.

    Here's what I think you meant.
    Let ##I_n = \int_0^{\pi/2} x \cos^n(x) dx##, with ##n \ge 0##
    i) Show that ##n I_n = (n - 1)I_{n-2} - n^{n - 1}##, for ##n \ge 2##.
    ii) Find the exact value of ##I_3##.
    Is this anywhere close to what you're asking?

    PS - I used LaTeX to format what I wrote. We have a tutorial here: https://www.physicsforums.com/help/latexhelp/. This is under the INFO menu, under Help/How-to.
     
  5. Apr 13, 2017 #4
    Sorry, I'm new here and have no idea how to format the text.
    But yes that's what I was trying to write , except part i) is i) Show that ##n I_n = (n - 1)I_{n-2} - n^{- 1}##, for ##n \ge 2##.
    Thanks for the help with formatting :)
     
  6. Apr 13, 2017 #5
    Apologies for that, here's the cleaned up, comprehensible version of the question (thanks to Mark44):
    Let ##I_n = \int_0^{\pi/2} x \cos^n(x) dx##, with ##n \ge 0##
    i) Show that ##n I_n = (n - 1)I_{n-2} - n^{- 1}##, for ##n \ge 2##.
    ii) Find the exact value of ##I_3##.
     
  7. Apr 13, 2017 #6

    Mark44

    Staff: Mentor

    I would try integration by parts twice, starting with ##u = \cos^n(x), dv = xdx##. After the second integration by parts, you should have an equation that you can solve algebraically for ##I_n## in terms of ##I_{n - 2}## and other terms.
     
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