University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A37H
Summer 2011
Assignment # 10
You are expected to work on this assignment prior to your tutorial during the week of
July 25th. You may ask questions about this assignment in that tutorial.
At the beginning of your TUTORIAL during the week of August 1st you need to submit
the following homework problems.
STUDY:
Improper Integral material  from LEC (this material is also found in the Chap
ter 14 end of chapter exercises), Chapter 22 (up to pg 457).
HOMEWORK PROBLEMS:
1. Determine whether the following improper integrals converge or diverge.
Find the
value of the improper integral if it converges.
(a)
Z
∞
∞
x
3
e

x
4
dx
(b)
Z
1
0
x
ln(
x
)
dx
(c)
Z
π
0
sec
2
(
x
)
dx
(d)
Z
∞
0
x
arctan(
x
)
(1 +
x
2
)
2
dx
Warning: You will need to use more than one integration technique to find the
appropriate primitive in part (d).
2. If the sequence
{
a
n
}
converges, then prove that its limit is unique.
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 Summer '11
 Katherine
 Math, Mathematical analysis, Limit of a sequence, Riemann integral, Henstock–Kurzweil integral

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