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MATH 31B  Lecture 4  Fall 2007
Solutions to Midterm 2  November 16, 2007
This is a closedbook and closednote examination.
Calculators are not allowed.
Please show all your work.
Use only the paper provided. You may write on the back if you need more
space, but clearly indicate this on the front.
There are 6 problems for a total of 100 points.
POINTS:
1.
2.
3.
4.
5.
6.
TOTAL:
1
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1. (15 points)
Evaluate the integral
Z
3
x
+ 1
x
4
+
x
3
+
x
2
dx
Solution.
Here we do a partial fraction decomposition and write:
3
x
+ 1
x
4
+
x
3
+
x
2
=
3
x
+ 1
x
2
(
x
2
+
x
+ 1)
=
A
x
+
B
x
2
+
Cx
+
D
x
2
+
x
+ 1
Comparing the numerators of both sides gives us:
Ax
(
x
2
+
x
+ 1) +
B
(
x
2
+
x
+ 1) + (
Cx
+
D
)(
x
2
) = 3
x
+ 1
Comparing the coeﬃcients for
x
3
,
x
2
,
x
, and the constants gives us the
following equations:
A
+
C
= 0
A
+
B
+
D
= 0
A
+
B
= 3
B
= 1
Solving the equations gives us
A
= 2,
B
= 1,
C
=

2,
D
=

3 Now we can
try to solve our integral.
Z
3
x
+ 1
x
4
+
x
3
+
x
2
dx
=
Z
2
x
dx
+
Z
1
x
2
dx
+
Z

2

3
x
x
2
+
x
+ 1
dx
= 2ln

x
 
1
/x
+
Z

2
x

3
x
2
+
x
+ 1
dx
Let us complete the square to evaluate the last integral. We let
u
= (
x
+1
/
2)
then
Z

2
x

3
x
2
+
x
+ 1
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This note was uploaded on 04/02/2008 for the course MATH 31B taught by Professor Valdimarsson during the Fall '08 term at UCLA.
 Fall '08
 VALDIMARSSON
 Math

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