Determine whether the integral converges or diverges, and find its value if it converges.

Answer

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: