CHAPTER 32
Improper Integrals
32.2
Determine whether J" (1 Ix
2 ) dx
32.3
For what values of p is J" (1 /x)
p dx convergent?
By Problem 32.1, we know that the integral is divergent when p = 1.
32.4
For p>l,
I In the last step, we used L'Hopital's rule to evaluate
32.5
For
convergent?
32.6
Evaluate £ xe~'dx.
260
32.1
Determine whether the area in the first quadrant under the curve y = l/x, for *£!, is finite.
This is equivalent to determining whether the improper integral J* (1 Ix) dx is convergent. J* (1 Ix) dx =
Thus, the integral diverges and the area is infinite.
converges.
Thus, the integral converges.
The last limit is l/(p-l) if p>l, and+=° if p 1.
is
First we evaluate J [(In x)/x
p ] dx by integration by parts. Let u = lnx, dv = (l/*
p ) dx, du = (\lx)dx.
Hence,
Thus,
Thus, the integral converges for all p > 1.
divergent for p :£ 1.
Hence,
for
by Problem 32.3. Hence,
is
is
By integration by parts, we find J xe * dx = -e *(x + 1) Hence, J
[In the last step, we used L'Hopital's rule to evaluate
dx convergent?