Question

How do you evaluate the integral `int 1/(x-1) dx` from 0 to 2 if it converges?

Answer 1

The integral does not converge.

Explanation:

The integrand is not continuous on the interval `[0,2]`. We try to find the improper integral by evaluating

`lim_(brarr1^-)int_0^b 1/(x-1) dx` and `lim_(ararr1^+)int_a^2 1/(x-1) dx`.

If these two improper integral converge, then `int_0^2 1/(x-1) dx` is equal to the sum of the two numbers.

Since `int 1/(x-1) dx = ln abs(x-1) +C`, neither integral converges.