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Evaluate the integral

  1. May 28, 2014 #1
    1. The problem statement, all variables and given/known data
    ∫ [0,∏/2] dx/[√sinx + √cosx]^4


    2. Relevant equations
    None


    3. The attempt at a solution
    ∫ [0,∏/2] dx/[√sinx + √cosx]^4
    = ∫ [0,∏/2] dx/[(√tanx + 1)^4 cosx^2]
    = ∫ [0,∏/2] sec^2(x) dx/[(√tanx + 1)^4]

    This is my attempt. I can only simplify. I'm stumped.
    Can you also give me advice on solving integral problems like this?
    Because it would be very nice to be able to evaluate problems like this without help.
     
  2. jcsd
  3. May 28, 2014 #2
    Try a u-sub.
     
  4. May 28, 2014 #3

    Ray Vickson

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    Maple manages to perform the indefinite integral, but the results are unenlightening: it is a 156-page expression involving complicated combinations of sin(x), cos(x), plus logarithms of such combinations, plus various Elliptic functions of such combinations. The definite integral is bordering on the un-doable, because of the need to evaluate that humongous expression numerically at the endpoints; just doing it numerically seems best. When letting Maple do it numerically (for increasingly many digits of accuracy) the given answer equals to the decimal representation of 1/3 to the given number of digits setting. So, it seems that the answer = 1/3 exactly! There absolutely must be an easy (or, at least, easier) way to see this, but so far I have not found one.

    For 20-digits in computations, Maple gets the answer as 0.33333333333333333333, while for 50-digits it gets 0.33333333333333333333333333333333333333333333333333, etc.
     
  5. May 28, 2014 #4
    Any chance you tried a u-sub?
     
  6. May 28, 2014 #5

    Ray Vickson

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    I don't know what Maple did to get the 156-page answer.
     
  7. May 28, 2014 #6

    Dick

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    Probably didn't think to pull out the sec^2 like Mathmanman and do the u-sub gopher_p suggested. Found a long way around.
     
  8. May 28, 2014 #7

    Ray Vickson

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    You should do an appropriate substitution if you think that a one-line result is better than a 156-page one.
     
  9. May 29, 2014 #8

    Dick

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    I did. u=tan(x). Isn't it better??
     
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