From the graph below, determine if each integral converges, diverges, or if its behavior cannot be determined.

(a)

(b)

(c)

(d)

(e)

(f)

Answer

(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

converges,

converges also.

(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

We know

converges, but since we don't know whether

diverges or not, we don't know what

does either.

(e) Since

converges and *g*(*x*) is greater than , we can't tell what

does.

(f) Since and

diverges,

diverges also.

The integral breaks up as

Since

diverges, so does the original integral.