Sec ~()V'\ s.;) : \3 5\ 5<0 BO , I ,

'Se(-·11 OY) 5.3; \\ 1 )0 I 35 I Lt \ I 'i fa c , 5, l{ ..:;:,ec1'1{)V) I \\, IS I ~\ I 3 \

t:. h.'Sec-ho'l J J B, 13 r; I 'Sec-hon J. \0 \3, rt, dqfd~1 3'-\,40, Lft, too, BS,8 Li, 9\

'Sec \i Dv'\ 3 Uu I~ 30 31 L{J 50I } f I ./ I

L 5 . .:( . \3

-The % Fvh o~ ~ 3ya~h (.\,'e 'SeM'.d .(\'s ­

=:0 --% + ~ 0\ vr :::. ~ ..,r -}- ].-rr.

;).

~l±1 () \ ~ S.::l \' ~,( J I ~(')(') d)(. = ,f.( x.') ct" 4' J.;l. tc",) dx.

-~1T(\r+ -k1[[.1)~

(d') j~ \~(x) \ Jx = S: \Hx} \'\)( ~ ~~ \ Hx} \ ~x

= ~ =P~1+)1T

5 . .:2.5\

):[H)(j -t 4JLl:)1J>. = j: )'("jch -I- ): l\3(xJd><

= \SW >.jJ", ~ '-\ ~5 3(";~x

= 5 -\- '-\ - \;}.

=@] Co.\c lA \ a \e. +hQ

J~ \("lJx::: \

).?'\ H x') ~ ~ ~\(xjAx - ~ ~ \'Cx') ctx

= s: H"jdx - [S: f(x;c\x - ~~ W,,)dllJ

~ 1- - l Li - I] == :+- 3

~\IJ

:TY"'I E'Xe\'r ;'S~5 \\ I :20, ~(J . in\e<JYo.\ LlSi.n3 fie T._

5.3.dO

5.3.35

5.3.LI\ In (xe.H··\ses 4\ a()('~ 4~ w~\e -+ht in\-e~yo \ a::;; a SUr(\ o-C. in tt?3va\s ~i~ou+ abso\uh: va \tVS

.o.l1d ?va \ lAale .

S_~ \x\ ctx = -x dx + S~ x dxS-:

5- L\ 5 1=i nc\ Gel) G' (0) aCId G' (ll/ Ll ) CA)ht.x~ ~ I I I

G("') = r ~a" -\: dt

,t=h's~ no-te ~a-\--

G (\) == r ~d dt =@] Nex.\-Qb~eK\IQ. -",a\- 'FTC :n:: 3ives

G \ Ci'.) = tCAI'\ X

. SD

5-4.\\

5_ Y. lB • Express, ~ o~-h-knvat1ve fCx') ~ {Cx) so\;sS,n3 ,~ 3\VeK\ ini+i~\ (Dod ,-hot') ex:. elY\ ·'(\~(3ro.\ '

~ (x) - e -~ F( Lf) := 0

. 1hi~ rY\£ o.t1S -\t,Glt I

we rlQQd -\0, "frl~ F.e JL .Jo~..l ,..J , F( x')::: ) : Vl t')dt = ): e t dt

. whiC\' amou",K -\u ~hJi"3 ..\\~ ~v.\iCtA\o."

·as

\.rS.4.3\.

-ro... S we hOVE:­

co",<f s _J 3(0'" s) (-s'r) S )

- sin s ( S(OS S - CDS <\ S )

5_ 5_ 8 \~ r€ teased wifh inita\mls _ \)~ ~ -to rf{\ lJ. \0­+0 c:k \-exrn I ni: ~ d-i;Joo (Q..

~IS-\- \S $_

L 5.5.13 uJhICh uJatef

\tl holf O"'l'u)kleft.- and ~-b, ~mO\J-(l t o~ 3 f'fitf\U+eS.

Our u,n'.\s ct -Hrne a"e YYI,nu+es) so uJe hove.

Let+ oW"'''ifY'()-hOf'\ '" '3 ~0 [ 50 + Y 6 -l % • 4 4 -l Y.:l + 4()]

- ±[;no] - l35 \i+erS

~i'3"'t- - ~i~- 3- 0 [ 41', t Lil. ~ 4'-1 t 4;) ~ 40 +sB]10

- +[ .2sej

a"~4:r;- oS· ~ lef+ - and ..13"'+ ­o.rfVl}~ i ma-\''()(\~ IS \3d- \; i-€-f!;. .

OI1J ('I'3h-\­os -l-ki hl,,~ SDmQ...

we. S'--'01\

III Exexr ,~" s '3 11 ..2l1 and::J{g) I I I

,HI +exmS ~ lA. onc\ d<.t .

r

I

t.H' u- l\x- \ ):X::::: Lf

~. :::- q =-) dx =- ~ cLu ~x 4

S/~l Jx= ~tL1:\) ~ -*Ju - ~y Sl \At 1Y' ..r.; ~ :: .l-- \ Iu:l -4- Ju + \) ~ ~

(Qq j\

L +C

d.u -= $(?c:Je =) de == dv de 'OL/e

. -S~ec7e)ew.e JG::o ~ (se/f}) e" - tG -:: Se "Au

:;:. e U -f C

to(1G (L-3]=e + ------.-._---­

~ := -'dx x

(in x)29~ () .;:l

- - - ~ du.-- ~ x~

JuJdu. =

- I)

~

C-- .- + 3

evaluate . l",dt~ Y\ \ \ <­

T-n EXeKC ,$.(' So

sx"( ).3 + I) ,A"

L.e.+ "':::)1.~t l 'So +\-)(\4­)

__du -::;. 3~~ =) ct>c:::: d,x

.Ilvte Ye~yt-3S~' (x3 ~ I')J)< '" j >\~ _ {} ~?

::: ; S (t\k , --I

-- -.3- -IJ -t C4

:= t~'+I~~ .. c ~\ s. (,.40

~ =..1)( =J Jx -:::: ck ~ 2x

~-~~=5 ;.; -~ ::: ~S -{; cW _ \ ,/h

:;l. . --rr;; -\: C

~+C

=~~'1 +C I

J " lxt I)'/~ Jx

lA::: x. -+ \ -) x:=: LA.. - \

+x " \ =) d4-'-':o ch

-S x (x-\ \)'14 C\X == ) (U - I) ,,."'1 J.u

= S(/1'1 - U'/'l) JIA

(O~ \;\. . :l~' du ~

::: ,;l. ~ cos I). Au

s~ ("4 I) l ~) -} ;) X ) ~ Jx

Le +- v..::".2 -+":Lx SO -thO.\­}M ~ .:I,)d; =-) d~ ~ Jt\ - _du ~ .,2,,-t;}< ;.(>\+ I)

~.l-(~-\.\\()(.;2-\-J).,)~ dX:= ,\"'-0) (M\> lJ."S. ~(' ~ J\) Ju.( I) .:l >1-\ I 1

~ S~ Y: JM

- (.ALI 1B 8 3

=: 51.).. El­f)

= , ~O~5 J

i ( a"] l=jt"J.. l~ eOi nh V ( ~) ~ 9(,\ ) .

)1;. ~.. 2. - ~~.

'-/- J. -I'" -_. -I "....

. (b) l=inct I o~ +

We h:Jv.e ~·l 5(~) > tlx} Ofl -d< ~ < ! I SD

. ~ O:(eo.. be +ween -th.t C\.ln.-t''S, Qr\ -\hIs 'Db\ '.s A := ~_~L(J ~~) - xJ dx

:= J- ~ (J - X - i\;L) Jx

~ ~dX - ~ - ·~11l ~. 3_~

~ U)~{- {)- (-LI~2+t)]

:OL:~(-'nJ

~. \ \Co =rn hex(i~e s 1\0 sha&d v-e5ion .

)S\("\X (OSX - Sln'X - ()

Sl (\ >< l;;leos x -- l) !::: 0

5in'X -::::-D . I

5.0 we. Y'<7"'C\ tD fblis a::. f.o"Q\U~ .'

~ 3tr -----'~--

.;l \\p

A- :::: A1-+ A;)- = (?.{3 - i)1r + ~ -- I - J!:­t.( co ;;;L 1(q

_ )11 f3 -1T -+ .Jl{..J3 -;;;; LI - 51f l_tS

_ )"11 [3 'r.) 4,,['3, -::>4 - ~_:rr }

{g. \. Sa . 'In (: x€Xr \";.es 3D 3+ o.nct 4.:> s\z.e-h: h ~. re3'Dt'"\I I

. e nclos.ed --rhe' CW('.(es Qf\d (O\'l\ f'V. toe. ,h OJea.b~ : as 0.Y'I inte3yal 0.\003 ~ >c Of ~ -o.~\.~ _

~ ,,2_Lo

J== I ~== to-x '3

/ ~ - (Ax'S

3 ;;;l.

x + x - \.:L - 0

0­ ca\cu\o-trK 1

~i2+J-~=O j:l- 'j _ (0 ::: 0

( j -+ 3) (~-) ) :=-() l ~ -- S)( ~-\-;»:::() ~/ ;X3; ~ ~'j¥-3; )~J ;;l

cj"ts

A- ~_:llio-{)- lj\1~

L A= ~_: [(Ic-tn- (-j)1~ +): I~-f) -~Jc1 = ~_~ (to- ~"1 J)J:j +r (lD- :r~)~ =L~J -~+il: ~ L~j -~-~ I c:: [ 0 - t l;> + ~+,))3+[(Id- - ~ - :l) -( 0 ) }

22 2J. - +-­-5 3

~ _I . Lf.).. . 'b \ /;;).. . \ x -} J =

We. (o.n \ira. ~C\.\-- 'k ~ CL.l\'tfi?S \I,,~eJ~q('l ·o~ (0, I) ard (\ I 0) O:Dc\ -\h(:\~ on 0< '" ~ \ I

· x+ ') == \ "es. above x I b ~ JIf"J = \ . 1\.'I\AS -\{1Q 0.1(0.

· ~.\Wet VI .\-\\i -hoo cur v Q~'<'J

A:0 )Jl \-,,) - ( \ --J~}<} dx

~)~ I \ -X- l \ - ~-r; h)~ ~" :c j~ l-)" U-J;) J~

;J L\ ~bJ I ::: [ -x -l 3 X ()

=(- \ 1 ~) - l 0 ~ 0) ~ GJ

* Tn ~ pn"L ~rl; LtV enco urI 1erY\,(l-'\ t C\

COmmon- -t:Jf.e. ,e \(l"e uJh,chI

+'Jt\c\ lnrnl \ j ~ lYloye

-the 5e.x\Q ( 0 , 50 k-t-t1 D (\

0\' Sl) \ ut It·~ '., . jf Yr~:rlhQ. r{)crhu: +nJ( ~ I JW

and becClMS~ 0..

+0 on:-W.e,r y()s.:}o~ ~.th~ fYl)blR m JUst

Co_ \ 50 +ht

«oyep'(()blcnlS o~ (Otn\T)CI' -f~ at rruthemdics- ,'j iv.rs~~ III os yy, ()ch 3eJ1QtO 1 ; ~ as

H\ fur r\ off?, VY'O.\us a (( fea \ - LUOylJ

a ~\ mplt: Llfflico 41on o~ -Jk 1noQ l S()IO+1C'n)

'j ~ ( '" ,.;;..

asks uS +0 +0 on .efl il ye

jOU ~ ( k ",J~ WI H ~ee -tho+

+\ii:;' <s:or+ an';e)

II

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