Determine whether the integral converges or diverges, and find its value if it converges.
This function is weird at x = 1. So we split the integral at x = 1.
Let's start with the right-most integral. When x is in the interval (1,2] the quantity x – 1 is positive, so we can drop the absolute value signs in the denominator.
As b approaches 1, the quantity b – 1 approaches 0 and so the quantity 2((b) – 1)1/2 approaches 0 also. This means we have a convergent integral:
Now for the other integral. Since x – 1 is negative when 0 < x < 1, we can get rid of the absolute value signs in the denominator by writing
|x – 1| = -(x – 1) = 1 – x.
We have to use substitution to find the antiderivative, since x has a coefficient of -1.
As b approaches 1, the quantity 2(1 – (b))1/2 approaches 0, so we're left with
Putting everything together, we see that our original integral converges: