Recently I took a quiz at my school, but the sentence I saw in the quiz really confused me. This is the problem that confused me (True or False)
So, I can prove that integral of sin(x)/x from 0 to infinity and -infinity to 0 are convergent with the Ratio Test. Thus, integral of sin(x)/x from negative infinity to infinity must exist by the improper integral's property. Therefore, there must exist the Principal value of the improper integral.
However, the reason this problem confused me is the term 'identity,' and 'between.' It looks like 0 to inf and -inf to 0 integrals are inverses to each other, so the identity must be 0 by abstract algebra. But, since they are all convergent integrals, isn't it more proper to say "The integral is the identity between convergent integrals?" I've never seen anything like this problem, so I assumed maybe this sentence will make sense to people who learned higher level of math.
Can I get help to interpret this sentence into a mathematical terms?
P.S. This is the other version of this problem which is also True/False
The answer for the first is True, and for the Second is False.
