# How do I find this integral?

Math9999

## Homework Statement

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

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 Helper

## Homework Statement

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

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 and member 587159
Math9999
I don't know anything about the integrand.

Dick
Homework Helper
I don't know anything about the integrand.
Do you know what even and odd functions are?

Math9999 and member 587159
Math9999
I know that the sine functions are odd, right?

Dick
Homework Helper
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 and member 587159
Math9999
That sin^7 (x) is also odd.

Dick
Homework Helper
That sin^7 (x) is also odd.

Math9999 and member 587159
Math9999
An even function?

Dick
Homework Helper
An even function?

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

Math9999 and member 587159
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 and Dick
Math9999
0?

Dick
Homework Helper
0?

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

ComplexVar89
member 587159
0?

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

Math9999