Lecture 6 Practice Exercises: Improper Integrals
At what points do the following integrals become improper?
R1
0
dx
√1−x2
R∞
0x3e−xdx
R∞
−1
dx
√|x|(x3+x−2)
R
π
2
0
sin(x)dx
√cos(x)
R5
0xe−xdx
Decide whether each of the above integrals converges or diverges.
Show that for xlarge enough
x3e−2x≤e−x
(Give an estimate at how large xmust be for the above inequality to hold.
Show that Z∞
0
x3e−2xdx
converges using the comparison test.
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