Determine if the integral converges or diverges. What integral are you using for comparison in each case?

(hint: for *x* > 1, we know sin *x* < *x* and )

Answer

Since the integrand looks sort of like , we're going to guess this integral diverges. To show divergence for certain, we need to find a function that is less than

whose integral diverges. Following the first hint, let's make the denominator bigger (and the function smaller) by replacing sin *x* with *x*:

Now following the second hint, make the denominator bigger again (and the function smaller again) by replacing square root with *x*:

We know that

diverges. Since

is a constant multiple of , this means

diverges also. Finally, we conclude

diverges.