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 )
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
is a constant multiple of , this means
diverges also. Finally, we conclude