p..
Spring 2012
MATH 1220-002: Exam 2
Name: ________________________
Robbie Sneliman
Instructions: The following questions are intended to assess your abilities on the basic concepts that we
have covered thus far. Answer all questions to the best of your ability and simplify all solutions as much as
possible. Regardless of the simplicity of the problem all work must be shown in order to receive full credit,
otherwise no credit will be awarded.
Problem 1: For this problem I
want
you to state the method you would use to solve the problem (DO
NOT solve the problem explicitly). The methods of integration we have covered are the following:
a.) U-substitution
b.) Integration by parts
c.) Partial fractions
d.) Trigonometric substitution
e.) Rationalizing substitution
i.
fxsin(x2)dx
U
~
2.
f~
3.
fexsin(x)dx
4.
f y2 ~4
—
y2dy
°~cT%L~cc~
i~c~h ~ ~
z f 1~2~dx
~.
ID
tD7~
~l5rob1em 2: Find the following limit or state that it does not exist,
tim tln(t)
~
t~t~
D
&~
—~
ID
o~s
Problem 3: Evaluate the given improper integral or state that the integral diverges (all work must be
shown to receive full credit),
~
Rrc~
(o~U~.
ç~
~_i,
I
~
~~ c c ~~
LI,~i2~
3
Problem 4: Compute the following integral,
/
sin5 (x)dx
~fl(~ c~
(
~
~
S(
—
-
-
~
~,
I
\~ç~
-2~ 2~. tu
L
4
-
~z~iL
z~it~zi
1
i’O
toe
Problem 5: Compute the blowing definite Integral,
p
J
0
ta
we
xsln(2x)dx
S ,s~tàIz,ô
J.Vt
~Ax
usã~s,
74m~~ ~
t~ft,sd
V
S
~
r
-
_____
4-
_____
4-
0
+.
C.
1-
It
S
4srnaô
IC
~çsh~L~ A%
a
____
___7_
~
S~’flhL~C)
4
II
0
5
,0
to
Problem 6: Compute the following indefinite integral,
I 2x2+x—4
I
dx
x3—.x-~—-2x
j
a
x(~_
2
-2~A
~
x
S
~
~
~
6
~
c
x-z~
Problem 7: Evaluate the following definite integral,
fe
I
j1 xln(x)
~
dx
c~w~v*~
~
~
1~
~1p~
~
~
~
~_~tt
=
J
X
~)
~
0
—~O
T~rw~1
~prc.pw
~
~
MU~1
hc
I ~
~
+
~
cy~x
A~rc~J
V.
IA
~Problem 8: Write down the definitions of the hyperbolic sine and hyperbolic cosine functions and use
these definitions to verify the identity cosh2(x)
sinh(x)
—
sinh2(x)
=
1.
cosh(x)
=
2-
Coc~N-
)
~
)
—
8
Extra CreditU~j~ts~’? Find the following
1. /
sin(2x)
d
2 j sin1(x) + cos4(x)
U~
~
(
~
~i-h~ac~
— C V ~(2
ç~Z1~~\~
c
~
J
I
-
~
((~ ~Cc~Qz~
~
I
•+~
CcS(~t~
(i~Ly~) + ~
____
S
3
~ ~Z~Jj~
~
9