> If the improper integral from -infinity to infinity is 0 then there t0 such that the integral from t0 to infinity equal minus the integral from -infinity to t0
If *f* is indeed integrable over **R** with (improper) integral 0, as is implicit in the statement of your exercise, then wouldn't it be the case that
- Integral\_-∞\^*t*\_0 *f*(*x*) d*x* = -Integral\_*t*\_0\^+∞ *f*(*x*) d*x*
for *every* *t*\_0 in **R**?
After all, when we take the equation
- Integral\_-∞\^+∞ *f*(*x*) d*x* = 0,
then decompose the domain (-∞,+∞) as the union (-∞,*t*\_0] ∪ [*t*\_0, +∞), what can we conclude?
Hope this helps point you in a useful direction. Good luck!