Integrability of sin(1/x) on [0,2]

[daniel_i_l]

Homework Statement

f is defined by: for x=<0 f(x) = sin(x). for x>0 f(x) = sin(1/x).
Is f integrable in [-2,2]?

Homework Equations

The Attempt at a Solution

I think that the answer is yes because f is continues for all x in [-2,2] except for a finite amount of points (x=0). Is that right? It just seems weird that a function as chaotic as sin(1/x) could be integrable around 0.
Thanks.

 

[CompuChip]

It looks like you are right.

As you already remarked, the problematic point is x = 0.
Actually, the question reduces to: is sin(1/x) integrable on [0, a] (with a > 0).
So is it integrable on [\epsilon, 2] and what happens if you take \epsilon \downarrow 0?