Calculus

We note that:

1. The function sin(x)/(1+x^2) has no singulariy between -1 and 1, so the definite integrals from -1 to 0 and 0 to 1 are defined and finite.
2. The numerator sin(x) is an odd function, while the numerator is an even function, which makes the integral from -1 to 0 equal and opposite in sign to that form 0 to 1.

From 1 and 2 above, we conclude that the given integral has a value of 0.

Note: I believe the word 'convergent' is used for the sum of a series, unless it is intended that the student breaks up the expression in a series.