If the integral above can be expressed as , where are all positive integers, find .
Notations:
Find the value of .
Let , and let denote the inverse function of .
Then the integral evaluates to where are rational numbers and is Euler's number.
If the sum can be expressed as where and are coprime positive integers, what is
The graph of (as shown above) looks innocent enough to noticeably oscillate as increases. However, as approaches , the oscillations grow rapidly, making vary greatly from around this region, even at very infinitesimal values of .
That said, will cross the -axis for an infinite number of times from to , creating several regions of the first quadrant enclosed by the curve and the -axis.
If the sum of these regions is , then determine .