did you know there's a formula for computing integrals of inverse functions? Think back: did this useful result appear in your basic calculus courses? 1/n

Probably not. How did this result somehow get weeded out of the curriculum while all these other Newton/Euler era integration techniques got passed down?

Guess what: IT'S A RELATIVELY RECENT RESULT. In full generality it first appeared publicly in 1965. 2/n

Even the special case dates from 1904?! Unbelievable. Nobody wrote down and published a proof until 1994!

There really is so much low hanging fruit in mathematics. You just need to be willing to ask silly questions. 3/3

Actually, one more thing. Looking at this formula, there must be some connection to integration by parts. We have a thing we want to integrate on the LHS, and on the RHS a product of an antiderivative (of dy) with the function, minus something computed by an integral. 1/n'

Now consider this tidbit from the wikipedia article (everything in this thread is from wikipedia):

2/n'

In research, one must scrutinize the literature mercilessly. We therefore type "integration by parts" into the wikipedia search bar and scroll until we see something familiar. And lo and behold:

3/n'

In other words, mod conditions, integrating the inverse of a function is integrating by parts! And if a function breaks up nicely enough into increasing and decreasing intervals, integration by parts should amount to using the inverse function integral on each interval! n'/n'

ok OK one more thing. I was flipping through random calculus results and guess what diagram showed up on the page for young's inequality (ie "the thing you need for Holder's inequality"):