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Homework Help: How do I find this integral?

  1. Dec 24, 2017 #1
    1. The problem statement, all variables and given/known data
    Find the integral of sin^7 x/(1+x^10) dx from -pi/2 to pi/2.

    2. Relevant equations
    None.

    3. The attempt at a solution
    sin^7 x means sinx to the 7th power. But how do I find this strange integral? I don't think u-substitution, trig identity, any of them will work.
     
  2. jcsd
  3. Dec 24, 2017 #2

    Dick

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    Think about symmetry. The interval is symmetric around the origin. What about the integrand?
     
  4. Dec 24, 2017 #3
    I don't know anything about the integrand.
     
  5. Dec 24, 2017 #4

    Dick

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    Do you know what even and odd functions are?
     
  6. Dec 24, 2017 #5
    I know that the sine functions are odd, right?
     
  7. Dec 24, 2017 #6

    Dick

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    Right. ##\sin(-x)=-\sin(x)##. What about the function you are integrating? What might that have to do with the value of the integral?
     
  8. Dec 24, 2017 #7
    That sin^7 (x) is also odd.
     
  9. Dec 24, 2017 #8

    Dick

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    Right. What about ##\frac{1}{1+x^{10}}##?
     
  10. Dec 24, 2017 #9
    An even function?
     
  11. Dec 24, 2017 #10

    Dick

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    You're 100% so far. Now what about their product? The function you are integrating?
     
  12. Dec 24, 2017 #11
    An odd function.
     
  13. Dec 24, 2017 #12

    Math_QED

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    Exactly. And what do you get when you integrate an odd function from -a to a?
     
  14. Dec 24, 2017 #13
  15. Dec 24, 2017 #14

    Dick

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    I'd feel better if you didn't end every statement with a '?'. Have some confidence!
     
  16. Dec 24, 2017 #15

    Math_QED

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    Yes, but why? Graphically, it is clear. Can you provide a simple proof?
     
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