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Sin and Cos Integrals

  1. Feb 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Compute [tex]\int^{\pi/2}_{0} \frac{sin^{2009}x}{sin^{2009}x + cos^{2009}x}[/tex]

    I used the identity [tex]cos^{2}= 1 - sin^{2}[/tex], but instead I set the exponent as 2009. And so I ended up with the answer being -1. I'm just wondering whether this is a legal solution or am I not allowed to do that. Thanks.
  2. jcsd
  3. Feb 2, 2009 #2
    [tex]sin^{2}x + cos^{2}x = 1[/tex] is the so called Pythagorean trigonometric identity. It is not valid when replacing the exponent 2 by another number, i.e.,

    [tex]sin^{n}x + cos^{n}x \neq 1[/tex] for [tex]n\neq 2[/tex].
  4. Feb 2, 2009 #3
    Thank you, I really needed that second opinion =)
  5. Feb 3, 2009 #4


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    Try the change of variables x -> pi/2-x to get a new integral. Then add it to the old integral.
  6. Feb 3, 2009 #5
    Thank you =) I found the answer.
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