11.4 Integral arithmetic
This technique goes by different names, but integral arithmetic captures the basic idea pretty well; sometimes it is very helpful to treat integrals as algebraic objects in order to find their value.
A very common example of this is where, by integrating (or any other integrable function) with respect to , we can arrive with an equation of the form (here we define to stand for \sayan integral we know to directly find the value of)
It is important that (because if is equal to one then we cannot solve for ), in which case we can just subtract from both sides, to solve for .
Find the value of
Solution: Start by integrating by parts (as in Section 11.2)
Then integrate by parts.
Overall then, we have
And we can add to both sides, giving that
and then after multiplying both sides by , we get that
Integrating by parts can get really messy - good presentation is key.