I was watching one of Leonard Susskind’s lectures on Satistical Mechanics and he mentioned an interesting way to integrate. He was deriving Stirling’s approximation for and had to evaluate
You can do it by parts of course letting and . Susskind observed that since the logarithm doesn’t vary much (not flat, but not much of a hike either) that for large values of you could treat it as a constant. The resulting integral is easy to do and you end up with as your anti derivative. But this isn’t quite correct as when you differentiate you obtain . But now its clear that if we had just added an to our anti derivative it would have come out okay, works.
Okay, can we try this on something else? What other functions do we know that are slowing changing for larger values of ? The inverse tangent function comes to mind. Again you can evaluate the following integral by parts,
but let’s just treat as a constant and then integrate. We’d obtain which upon differentiating gives
which isn’t quite right, but we can see how to correct it by subtracting the term .
How often is this useful? I’m not sure. What’s nice is that it makes use of the behavior of the function and the corrections (at least in these two examples) are easier to figure out than the original integral. In the case where we’re doing very general integration perhaps this could be of help?
NOTE: After a bit more thinking this does boil down to integration by parts. Consider an integral like
That you don’t know the antiderivative to but suppose is relatively slow changing over a large chunk of its domain (I know rigorous isn’t it). Then
To check, differentiate,
So you would need to adjust the initial ‘guess’ to
Which is exactly integration by parts. Though integration by parts works in cases where you don’t need to assume such things about the function.