I don't understand the answer to the following definite integral. Can anyone help me explain why it is the way it is?

I think the easiest way to see my point on this is to use u-substitution (though it isn't necessary):

And I imagine going from the last step to the next one is my problem....

But the answer cannot be 0! If you think about this being an area-under-the-curve problem, then the area is infinite because of the asymptote at the discontinuity within the limits of integration.

My suspicion is that

is somehow imaginary, but if it is I don't know why.

Can anyone explain this in reasonably understandable terms?