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Chain rule integration help

  1. May 19, 2007 #1
    1. The problem statement, all variables and given/known data

    Find the integral of (Sin^4(x)*Cos^4(x)) in respect to x without using a calculator.

    2. Relevant equations

    Sin^2(x) + Cos^2(x) = 1
    Sin^2(x) = (1-Sin(2X))/2
    Cos^2(x) = (1+Sin(2X))/2
    Sin(2x) = 2Sin(x)Cos(x)
    Cos(2x) = Cos^2(x) - Sin^2(x)

    3. The attempt at a solution

    I've been attempting to manipulate the equation so-as-to isolate a chain rule factor so I can use the U subsitution method on the problem. Unfortunately, my efforts so far have been fruitless.
     
  2. jcsd
  3. May 19, 2007 #2
    Unfortunately these are both even powers, so you won't be able to get a u-substitution out of it (at least not a trivial one). You can do some sneaky algebra to get it into an easily integrable form, or just put it all in terms of sine or cosine to a power, and use a reduction formula.
     
  4. May 19, 2007 #3

    malawi_glenn

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    Sin^4(x)*Cos^4(x) = (sin(x)cos(x))^4 = ((1/2)Sin(2x))^4

    than i would use Eulers formulas:

    sin(b) = (e^ib-e^-ib)/(2i)

    cos(b) = (e^ib+e^-ib)/(2)

    Have also tried using Eulers formulas?
    My favorite in integrating trigonometric functions.
     
  5. May 19, 2007 #4

    malawi_glenn

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    so you can express the Sin^4(x)*Cos^4(x) as

    (-1/16)*(2cos8x - 8cos4x + 6)

    can have done some errors, it is very late now in sweden hehe
     
  6. May 19, 2007 #5
    I haven't tried the Euler's forumula method, though I doubt that is what is intended.

    In class, the answer to that and several other similar problems were put up in random order to show where we should be at the end. The ultimate answer was quite clean, just a sum or difference of trigonometric fractions of different powers. I don't quite remember the exact answer, though I believe the denominators were 128.

    Thank you all for your help in the meantime.
     
  7. May 19, 2007 #6

    malawi_glenn

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    {should be (-1/16^2)*(2cos8x - 8cos4x + 6)

    yes:)

    Eulers formula ALWAYS work, i have never seen a trigonometric function that it dont works on. You dont have to remember identities and so on, just Euelers two formulas hehe
     
  8. May 19, 2007 #7
    Thank you for your help Malawi, that's a start for me, though I don't believe that's the answer my teacher is looking for. I'll try manipulating the Euler's formula some more.
     
  9. May 19, 2007 #8

    malawi_glenn

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    No it is not the answer, i just said that you can express (sin^4(x)cos^4(x)) as that..

    and yeah, forget the (-1/16^2) , shoulb be (1/16^2) = 1/256
     
    Last edited: May 19, 2007
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