Say we want to find . Normally, we let , or realise that the integrand is the derivative of , or do something along similar lines, but suppose we ignore those. Instead, we do the following:
Write and let:
More generally, letting gives:
My Question (1): why are these identities not used and instead the arctangent ones are? I even suspected that I was missing something, but differentiating quickly confirmed that they are indeed correct.
We know that
But using the relation found above gives:
, which doesn't converge, since the logarithmic function is not defined on .
Question (2): why is this? I probably need to pick up a complex analysis book!