How to Prove the Integral of sin^7 x/(1+x^10) dx from -pi/2 to pi/2 is 0?

[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.

 

[Dick]

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?

 

[Math9999]

I don't know anything about the integrand.

 

[Dick]

I don't know anything about the integrand.

Do you know what even and odd functions are?

 

[Math9999]

I know that the sine functions are odd, right?

 

[Dick]

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?

 

[Math9999]

That sin^7 (x) is also odd.

 

[Dick]

That sin^7 (x) is also odd.

Right. What about $\frac{1}{1+x^{10}}$?

 

[Math9999]

An even function?

 

[Dick]

An even function?

You're 100% so far. Now what about their product? The function you are integrating?

 

[Math9999]

An odd function.

 

[member 587159]

An odd function.

Exactly. And what do you get when you integrate an odd function from -a to a?

 

[Math9999]

0?

 

[Dick]

0?

I'd feel better if you didn't end every statement with a '?'. Have some confidence!

 

[member 587159]

0?

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