Evaluate the following integrals. Show your work'
1)
alffi*
X+I
T
r [--dxr8 *d:&f*
n,F,,u
flu-r
1a
fx*r)e
t;alJ' x-r
{ X*f
r
"
k
(v-f\hI
-j
,,
|
--A
d ..* Al v- l -l L
I
(
ry,
h*\13*X'o
€-
als o
XoW
rl
4.;k
* ]r{
t =L
tr
Lttt
4
3J e u '& , =
,l
Q*hL
qt* tLi
b=e
rh
b
\
Sa !a_
{uou&
" -A
au_kaL- 6eqq *6 {nVq =3uQr?-6"?u'+{nq<
( =eq
"*n""{
6*Z
(Hint: Start by substitutin1 +F).
.wd,,
d^*i
=1rt4*I
e
r
f (g*td'n 4 +-L
= o,(sut--6nnq,6)u-3
fT
c) | -7dt
J r Vr -
--
t
,l.T
r"=Ux-i y=4t1.I
t
\r_
J
,(t" =
hql-t
,l
d'r-***
d+
f
ur
ZIIxT-
€
(*
l
*
c
q@b,*,
((d+a
*rE
a>l&*
r=o*
tI t&-t:e,hs.
I#nFr.
*ffi
J*- se*,9"tzL
*s
o*,ed*- \*.A*
: ( ,tt',L,g-3 Jl{*gl
ffi-
' 5ffi---
'\ vhe'
-;i4
rn#' I
#l
fffl*e
z (,r- * + C
:-
_ _L +Wo
tIr_i'-
fifur*1*n'
E"grt'[*cr=xa+f
'*& "*o
t'c
=*:*t-a-(-
f lffi
P& k7*,*4
2) For each of the following improper integrals determine, using a suitable comparisontest, whether
the integral convergesor diverges. (You do not need to calculate the values of convergentintegrals.)
State clearly what compa,risonyou axeusing, what function you compare with, and show that f < g,
respectivelywork out the limit of f I g. No points wilt be given if no work is shown,or if only an argument
of polynomial degreeswithout explicit comparisonis done.
il|,*ffi*
N
^
Ne-$il itile\
lrhu
A-L co"ry<r,\ oo&" Jf
Ll^
./
,r&+uu.r
/#-
- a/
\av
-"frsr-qe,
fv.
CFe
T%'&,
tt
= I
G*
{#{,"W4
* \fi
I
*l
I
\a
#tt
{+a
-''L.
Yd{sr*ff;
('/,-t),fuk
,2x = A, il
ir+3
It
;t;fl
I
f{ery}qrffi "tg*t
!
Y4F3k{.((
o)f, t*a*
(^'4* f)
d
A-tny&{**/6
,h^,G-'6*
S
il*[*r'*r*N. ,€* #
$h"
=L,WtL"
/#J]
= ,('
I+p
+/,fu=fu f7uaq)
f"tt(-)
rotqx, fu,lq d-*;de*son**nl*n
Mfr*4u
J ft^a"
/=
fna
{
i
E
I
$*
ft&.,ry"sdr.
<-E-*
X
3) The following multiple choiceproblemswill be gradedcorrect answeronly. You do not need to
show work, and no partial credit will be given. Record your answerin the answerblock on the front
page. Answersglven on these pageswill not be scored. You also may tear off these pagesand do not
needto hand them in.
It is strongly recommendedthat you work out the problems until the correct answer is uniquely
*intuition" or
"guessing"- doing so is likely to result in
determined and don't just try to solvethem by
a wrong pick.
Each correct arrwer is worth 6 points, each incorrect amsweris counted as 0 points. (Unanswered
questions are L point, questionsin which more than one allswer is ticked are consideredto have been
answeredwrongly.)
What islim".-o
ffil
A)
E-1
B)
Eo
tr'@3 E
The limit doesnot exist
Integrating / 2r arcsin(n)dnby parts, we obtain an expressionof the form
A - I Bdr. What is B?
E
C)
12
t-rz
2r
,/L - rz
E
/ ry z \
Il Y, ll. |
arcsin(r)12
1
tr G712
haslimit L - 3. Determinethe
an:ry,which
Considerthe sequence
smallest valueof .lf , for which lon-:1":4m
/-
for any n > N.
E25E1ooE2ootr3ooq1p
e,(
D)
Solvethe integral [ ,
?'
,, O,
J (u + 2)(n+ I)**
-L
xpc_,W'{
9 ;*Ft+*E"til*ls""f"ll+ c _F 2tnlrl+ ?*+ |rz+ c
h
i"
+n
+
E
lr
J
l,i t1+c
S$i{q-ii-trf.tL*T'
H
E)
Let n be a number for which the improper integral
f*
dr
J. rus;t;
converges.Determine the value of the integral
1
L /"--* \
log n
ri log n
lal
rel
n* L
Lur
; (tr;J
tsiffi
n_r
\_/
F)
Which of the followingsumsdoesnot equal
481632
r.4 +u;+2.8+g*+"'
r
n:l
oo
E
2 .2"(2n)l
( 2 n +2)l
2n
r
oo
t hl
rv l
T
/, (zn)(2n+ L)
?
2n*r
rd $2"(zn-z)r
+2) u 3
H Q" + L)(2n
Qn)t
rel Z-J
Y
"1(2"
- L)(2n)
If you are done and have time left, check your answers
on all the problems. Is in each problem
clear, what your answer is? Did you tick the correct boxes
on the multiple choice questions? Do your
calculated antiderivatives differentiate correctly to the original
function?