1
Math 20B Midterm 2
Review Outline
Basic Information for the Midterm Exam:
The
midterm is cumulative but will have emphasis on Sections 7.17.4, 7.7, 7.8,
Supplement 2.2, 3.4, 4.6, 11.111.4
; it will not focus too much on any particular topic.
However, in a math class, it all builds upon itself.
You will
not
be allowed a calculator on the exam, so please do not bring one.
You
should
bring a number two pencil. (You can bring more than one if you feel so
inclined.) You are permitted a
handwritten
reference sheet on the exam (8.5 x 11
inches). You can put whatever you feel is important on it (see the rest of this document
for ideas.) Please,
do not bring anything more than this.
We reserve the right to place your backpacks in the front of the class. You don’t need to
worry about bringing a blue book, as you will be able to write directly on the exam.
The exam will be held on
Friday, November 16
th
, 2007
, in class (
LEDDEN AUD
).
You should have sufficient time to go back through your work and check your math.
Remember,
does your answer make sense?
(Draw a picture/plug numbers in.)
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Math 20B Midterm 2 Review Outline
2
Section 7.1: Integration By Parts
Know how to integrate using integration by parts
The formula is
udv
uv
vdu
=
−
∫
∫
.
Know how to choose
u
and
dv
(following LIPET)
 see the chart to the right
 ex. (i)
4
0
sin(2 )
x
x dx
π
∫
; (ii)
16
1
ln( )
x
x dx
∫
; (iii)
3
1
ln( )
e
x
x dx
∫
;
(iv)
1
3
0
z
ze
dz
−
∫
; (v)
2
1
3
0
x
x e
dx
∫
; (vi)
3
2
2
0
cos(
)
x
x
dx
π
∫
 ex. Evaluate
3/2
ln( )
x
x dx
∫
;
ln( )
x
dx
x
∫
;
2
sin( )cos ( )
x
x
x dx
∫
Section 7.2: Trigonometric Integrals
Know how to use integration by parts to find integrals of sin
n
(
x
) and cos
n
(
x
)
Know the strategy for evaluating sin
m
(
x
)cos
n
(
x
) for different combinations of
m
and
n
Know the trig identities
2
1
cos(2 )
cos ( )
2
x
x
+
=
and
2
1
cos(2 )
sin ( )
2
x
x
−
=
Know how to do something similar for tan
m
(
x
)sec
n
(
x
)
 ex. Evaluate
4
4
4
0
sec
tan
d
π
θ
θ θ
∫
;
3
5
sin ( )cos ( )
x
x dx
∫
;
11
2
0
cos( )sin
( )
x
x dx
π
∫
 ex. Find the average value of
2
( )
1
f x
x
=
−
on [1, 1].
Know the sumtoproduct and producttosum formulas for sine and cosine

[
]
1
2
sin
cos
sin(
)
sin(
)
A
B
A
B
A
B
=
−
+
+

[
]
1
2
sin
sin
cos(
)
cos(
)
A
B
A
B
A
B
=
−
−
+

[
]
1
2
cos
cos
cos(
)
cos(
)
A
B
A
B
A
B
=
−
+
+
Supplement 2.2: Integrating Products of Sines, Cosines and Exponentials
Know how to use the formulas for sine and cosine from Supplement 1.2 to express
sine and cosine in terms of exponentials
 ex. Evaluate
2
cos( )
i
x
e
x dx
∫
;
2
sin(4 )
i
x
e
x dx
−
∫
;
5
cos(3 )
i
x
e
x dx
∫
 ex. Use complex exponentials to evaluate
cos(2 )sin(7 )
x
x dx
∫
and write the
result in terms of trigonometric functions.
 ex. Evaluate
5
5
(
)
i
x
ix
i e
e
dx
−
−
∫
and
(
)
ix
ix
e
e
dx
−
+
∫
; write the result using only
realvalued functions.
u
dv
L
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