# 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