Evaluate the integral [-1,1] sinx / 1+x^2?

Question

Somehow, you need to use substitution - I think arctanx should come into it somehow...

Answer 1

Proof of why this definite integral is 0:

Let x be negative -x. Then we have

Sin(-x) / 1+(-x)^2 = -Sin(x) / 1+x^2 = - ( Sin(x) / 1+x^2 )

That proves that it has reflexive symmetry about the origin (0,0), so that the area from x = 0 to -1 is negative of the area from x = 0 to 1. Hence, the total 0.

I don't even have to bother finding the indefinite integral.

Answer 2

Good thinking, but I'm not seeing it.

If y = arctan x, we have the integral of sin(tan(y))dy

That would have to be some kind of chain rule special with convenient cancellations, I would think, but nothing is jumping to mind.

EDIT: Ooh. Caught me napping. Yeah. It's zero.