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Integral help

  1. Feb 27, 2010 #1
    1. The problem statement, all variables and given/known data
    [tex]\int sin^{2} u - cos^{2} u / \sqrt{sin^{4} u + cos^{4} }[/tex]

    2. Relevant equations

    3. The attempt at a solution

    [tex]\int sin^{2}(u) - cos^{2}(u) / \sqrt{sin^{4}(u) + cos^{4}(u)}[/tex]
    \sqrt{sin^{4} u + cos^{4}}
    = \sqrt{(sin^{2}(u) + cos^{2}(u))^{2} - sin^{2}(u)cos^{2}(u)}
    = \sqrt{1 - 2sin^{2}(u)cos^{2}(u)}
    = \sqrt{\frac{1+cos^{2}(2u)}{2}}
    OR \sqrt{\frac{2 - 2sin^{2}(2u)}{2}}[/tex]

    that's as much as I could simplify.. any help would be appreciated, thanks
    Last edited: Feb 27, 2010
  2. jcsd
  3. Feb 27, 2010 #2


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    Science Advisor
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    Your numerator is pretty close to being cos(2u). Can you express the denominator in terms of sin(2u)? Then you should be able to see the substitution to use.
  4. Feb 27, 2010 #3
    wow, the substitution has been right in front me! I've been blind..

    so the denominator is [tex] \sqrt{\frac{2 - sin^{2}(2u)}{2}} [/tex]

    and the numerator is -cos(2u)

    so I can use t = sin(2u)

    Last edited: Feb 27, 2010
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