How do I find this integral?

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  • #1
Math9999

Homework Statement


Find the integral of sin^7 x/(1+x^10) dx from -pi/2 to pi/2.

Homework Equations


None.

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.
 

Answers and Replies

  • #2
Dick
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Homework Statement


Find the integral of sin^7 x/(1+x^10) dx from -pi/2 to pi/2.

Homework Equations


None.

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.

Think about symmetry. The interval is symmetric around the origin. What about the integrand?
 
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  • #3
Math9999
I don't know anything about the integrand.
 
  • #4
Dick
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I don't know anything about the integrand.
Do you know what even and odd functions are?
 
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  • #5
Math9999
I know that the sine functions are odd, right?
 
  • #6
Dick
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I know that the sine functions are odd, right?

Right. ##\sin(-x)=-\sin(x)##. What about the function you are integrating? What might that have to do with the value of the integral?
 
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  • #7
Math9999
That sin^7 (x) is also odd.
 
  • #8
Dick
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That sin^7 (x) is also odd.

Right. What about ##\frac{1}{1+x^{10}}##?
 
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  • #9
Math9999
An even function?
 
  • #10
Dick
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An even function?

You're 100% so far. Now what about their product? The function you are integrating?
 
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  • #11
Math9999
An odd function.
 
  • #12
member 587159
An odd function.
Exactly. And what do you get when you integrate an odd function from -a to a?
 
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  • #13
Math9999
0?
 
  • #14
Dick
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0?

I'd feel better if you didn't end every statement with a '?'. Have some confidence!
 
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  • #15
member 587159
0?

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

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