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How to start with integral

  1. Mar 28, 2009 #1
    1. The problem statement, all variables and given/known data

    Evaluate:

    [tex]

    \lim_{n\to\infty} \int_0^{\frac{\pi}{2}} x|\cos nx|\ \mathrm{d}x

    [/tex]


    2. Relevant equations
    hint: the integral is not zero.



    3. The attempt at a solution
    I don't know how to start: how do I deal with the absolute sign?
     
  2. jcsd
  3. Mar 28, 2009 #2
    Re: integral

    make two cases. One for the positive and the other for the negative
     
  4. Mar 28, 2009 #3
    Re: integral

    How do I know where it becomes negative?
     
  5. Mar 28, 2009 #4
    Re: integral

    1) all positive terms

    2) bring out the negative outside of the integral.
     
  6. Mar 28, 2009 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: integral

    It changes sign everywhere cos(nx) vanishes, when nx is an odd multiple of pi/2. You might find it easier to count if you do the change of variables u=nx first. Then follow tnutty's advice and add up the positive parts and negative parts separately. Try and guess the answer first. For large n you get many cosine cycles. So it ought to be the integral from 0 to pi/2 of x*(the average value of |cos|).
     
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