# How do I find this integral?

1. Dec 24, 2017

### Math9999

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. Dec 24, 2017

### Dick

Think about symmetry. The interval is symmetric around the origin. What about the integrand?

3. Dec 24, 2017

### Math9999

I don't know anything about the integrand.

4. Dec 24, 2017

### Dick

Do you know what even and odd functions are?

5. Dec 24, 2017

### Math9999

I know that the sine functions are odd, right?

6. Dec 24, 2017

### Dick

Right. $\sin(-x)=-\sin(x)$. What about the function you are integrating? What might that have to do with the value of the integral?

7. Dec 24, 2017

### Math9999

That sin^7 (x) is also odd.

8. Dec 24, 2017

### Dick

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

9. Dec 24, 2017

### Math9999

An even function?

10. Dec 24, 2017

### Dick

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

11. Dec 24, 2017

### Math9999

An odd function.

12. Dec 24, 2017

### Math_QED

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

13. Dec 24, 2017

### Math9999

0?

14. Dec 24, 2017

### Dick

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

15. Dec 24, 2017

### Math_QED

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