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MAT1332
Spring/Summer 2009
Assignment 2 Solutions
1. Long division gives
6
x
3

4
x
+
8
3
x
2

9
x
=
2
x
+
6
+
50
x
+
8
3
x
2

9
x
.
The denominator has two distinct real roots and in particular 3
x
2

9
x
=
3
x
(
x

3). We
would like to ﬁnd
A
and
B
such that
50
x
+
8
3
x
2

9
x
=
A
3
x
+
B
(
x

3)
. Taking
x
:
=
0 and
x
:
=
3 gives
A
=
8
3
and
B
=
158
9
. Thus,
Z
6
x
3

4
x
+
8
3
x
2

9
x
dx
=
x
2
+
6
x

8
9
Z
1
x
dx
+
158
9
Z
1
x

3
dx
+
C
=
x
2
+
6
x

8
9
ln

x
+
158
9
ln

x

3
+
C
,
and we conclude that
Z
2
1
6
x
3

4
x
+
8
3
x
2

9
x
dx
=
9

1
9
(
8ln

2
+
158ln

2

)
=
9

166
9
ln

2

which is
≈
3.7847.
2. The integral converges. This can be seen using the comparison test. First, observe that,
for any
x
,
x
6
<
x
6
+
1,
whence
1
1
+
x
6
<
1
x
6
.
As such, since
R
∞
1
1
x
6
dx
converges to
1
5
it follows that
R
∞
1
1
1
+
x
6
dx
converges.
3. First, note that lim
x
→
0
x
2
=
0
=
lim
x
→
0
1

cos(
x
). Thus, we may apply l’Hôpital’s rule to
conclude that
lim
x
→
0
1
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 Fall '07
 MUNTEANU
 Calculus, Division

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