CK-12-Calculus

7.6. Improper Integrals http://www.ck12.org

∫+∞

0

dx

ex+e−x=llim→∞

∫ 1

0

dx

ex+e−x

=llim→∞[tan−^1 ex]l 0

=llim→∞

[

tan−^1 el−tan−^11

]

=π 2 −π 4 =π 4.

Thus the integral converges to

∫+∞

−∞

dx

ex+e−x=

π

4 +

π

4 =

π

2.

Multimedia Links

For a video presentation of Improper Integrals(22.0), see Improper Integrals, http://www.justmathtutoring.com (6:23).

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/610

For a video presentation of Improper Integrals with Infinity in the Upper and Lower Limits(22.0), see Improper Int
egrals, http://www.justmathtutoring.com (7:55).

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/611

Review Questions

1. Determine whether the following integrals are improper. If so, explain why.
   a.∫ 17 xx−+^23 dx
   b.∫ 17 xx++^23 dx
   c.∫ 01 lnxdx
   d.∫ 0 ∞√x^1 − 2 dx