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!