Improper Integrals

Question

Evaluate the improper integral or state that it diverges.

lower bound: -2

upper bound: 3

f(x) = (1/(x+1))

I know that the integral converges to ln(4), but I would like to receive an explanation to one of the later steps to solving this problem, namely the following:

limit of ln(|b+1|) as b approaches negative one from the LEFT. Why is the limit of this -infinity?

Answer 1

I would argue as follows:

f(x)=1/(1+x) is antisymmetric about x=-1, that is f(-1-h) = -f(-1+h)

Therefore the integral of f(x) from -2 to 0 is algebraically 0....despite the fact that the function is unbounded on that interval. I am not sure that this argument "holds water", that is, I don't know how to prove this analytically.

However, if this is true, then the integral you want is simply the integral from 0 to 3 of f(x) which is 4.