From the graph below, determine if each integral converges, diverges, or if its behavior cannot be determined.
(a) The function is the same as the function . We know the integral of this function from 0 to 1 converges, and the integral of this function from 1 to ∞ diverges.
(b) Since and
(c) We don't know what the integral
does. We know that
diverges, but since that doesn't help us at all.
(d) We can break up this integral as
converges, but since we don't know whether
diverges or not, we don't know what
converges and g(x) is greater than , we can't tell what
(f) Since and
The integral breaks up as
diverges, so does the original integral.