Integral Calculus - Problem Set
September 2, 2022 | Author: Anonymous | Category: N/A
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YJKEX @J^[‛Y JELGMY IAMMGLG AC RJ@RJELJ IAMMGLG AC GELKEGG^KEL IKSKM GELKEGG^KEL HGRJ^X@GEX @JXFG@JXKIY, YW^SG[KEL YW^SG[KEL JEH X^JEYRA^XJXKAE X^JEYRA^XJXKAE GELKEG GELKEGG^KEL G^KEL
KEXGL^JXKAE CA^@WMJY Jmlgnrjki, Gxpa Gx paeg eget etkj kjm, m, Maljrktf`ki Cueitkaes
:.
∯ j hu0 j∯ hu0 ju +I
>.
∯ j hu0
jeh jeh
u
u
j
+I , j5 :, j ≢ :
me j :
car e≢ ∞: ∯ u hu 0 e +: u + + I car e
9.
e
:
∯ g hu0g +I hu ∞ 7. ∯ u hu 0∯ 0 me|u|+ I u u
8.
u
:
Xrklaea`gtrki Cueitkaes
3.
∯ me u hu0u me|u|∞u +I
. ∯ isiuhu 0me|isiu ∞iat u|+ I 6.
± ∞me|isiu + iat u|+ I
∯ sgiu tje u hu0 sgiu +I
:9.
:8.
isiu iat u hu 0∞isiu + I
∯ sgi ¾ u hu0 tje u +I ∯ :3.∯ isi ¾ u hu 0∞iat u + I :7.
Kevgrsg Xrklaea`gtrki Cueitkaes
∯
:
hu
u j
0jriske + I 0
:
∔ j + u ¾ j >
:
hu
∔
>
u u ∞j
jritje
u + I j u j
0 jrisgi + I ¾
j
YJKEX @J^[‛Y JELGMY IAMMGLG AC RJ@RJELJ IAMMGLG AC GELKEGG^KEL IKSKM GELKEGG^KEL HGRJ^X@GEX
∯ jriske u hu0 u jriske u +∔ :∞u +I >:.∯ jritje u hu0u jritje u∞ me ∔ : + u ¾ >
>2.
Fypgrnamki Cueitkaes
∯ skef u hu 0iasf u +I >9. iasf u hu 0 skef u + I ∯ >8.∯ tjef u hu0 me iasf iasf u + I >7.∯ iatf u hu 0 me|skef u|+ I >3.∯ sgi u hu 0 tjef u + I >6.∯ sgifu tjef u hu 0∞ sgifu + I >;.∯ isifu iatf u hu0∞ isifu + I
>>.
>
∯ ∔ uhu+j 0skef∞ u z +I :
92.
>
∯
9:.
>
hu
∔ u ∞j ¾ >
0iasf∞
:
u + I , u 5 j 5 2 j
: u 0 tjef∞ + I ,,uu = j ¾ ∯ j hu j ∞u ¾ j :
9>.
>
>
∞: u 0 iatf ∞ + I , x 5 j ¾ ∯ j hu j ∞u ¾ j :
99.
Atfgr Cueitkaes
>
>
hu
98.
0 me|u + ∔ u ¾ º j ¾|+ I
∯ ∔ u º j ¾ : j +u 0 me | +I , u = j ¾ ∯ j hu ∞u > j j∞u | >
>
97.
>
>
∯ u ∞huj ¾ 0 > :j me |uu∞+jj|+ I , u 5j ¾ >
93.
>
∯ ∔ j ∞u hu0 u> ∔ j ∞u + j>¾ jriske uj +I >
9
>
>
∯ ∔ u ¾ º j ¾ hu 0 u> ∔ u º j + j>¾ me|u +∔ u ¾ º j ¾|+I >
96.
Ketglrjtkae ny
∯ uhv0 uv ∞∯ v hu
9;.
>
YJKEX @J^[‛Y JELGMY IAMMGLG AC RJ@RJELJ IAMMGLG AC GELKEGG^KEL IKSKM GELKEGG^KEL HGRJ^X@GEX
Rjrts Xrklaea`gtrki
Ya`g ketglrjtkaes `jy ng sk`pmkckgh wktf tfg cammawkel sunstktutkaes4
Yunstktutkae
:. Kc j je e ketglrje ketglrjeh h ia iaetjke etjke
∔ j ∞ x , >
sunstktutg x >0 j sk ske e¾ζ >. Kc j je e ketglrje ketglrjeh h ia iaetjke etjke
∔ j + x ¾,
Yunstktutg x >0 j tje¾ ζ 9. Kc j je e ketglrje ketglrjeh h ia iaetjke etjke
∔ xx ∞j, >
Yunstktutg x >0 j sgi ¾ ζ @arg lgegrjmmy, je ketglrjeh tfjt iaetjkes aeg ac tfg car`s
∔ j ∞n x , ∔ j +nx ¾ ,∦ ∔ n x ∞j >
>
nut ea atfgr krrjtkaejm cjitar `jy ng trjescar`gh keta jeatfgr kevamvkel trklaea`gtrki cueitkaes ac j egw vjrkjnmgs cammaws4
Car
Wsg
j sk ske e¾ζ n j > tje ¾ ζ ∔ j +nx ¾ x 0 n tje j > ∔ n x >∞j x 0 n sgi ¾ ζ
∔ j ∞nx ¾
>
x0
Xa antjke
∔ j ∔ :∞ske¾ ζ 0∔ j ias ζ
tje e ¾ ζ 0 ∔ jsgiζ j sgiζ ∔ j ∔ :+ tj
∔ j ∔ ssgg i ζ ∞:0∔ j tje ζ >
ς //>
\jmmks Car`umj
∯ ske
`
e
ζ ias ζhζ 0± ± ±
2
wfgrg4
RMJEG J^GJY
κ 0 ς / > wfge natf ` jeh
κ 0: kc atfgrwksg
` ∬e 0± pasktkvg ketglgr, eat gqujm ta
Wskel Sgrtkijm Ytrkp
e
jrg gvge
:
YJKEX @J^[‛Y JELGMY IAMMGLG AC RJ@RJELJ IAMMGLG AC GELKEGG^KEL IKSKM GELKEGG^KEL HGRJ^X@GEX x>
∯ ( [ ∞[ ) hx
J 0
W
M
x:
[ W 0 c (( x ) ? [ M0 l ( x ) [ W 0 l ( x ) ?[ M0c (( x )
Wskel Farkzaetjm Ytrkp
[ >
∯ ( V ∞ V ) hy ± ±
J 0
^
M
[ :
V ^0 l ( y ) ? V M 0c ( y )
Ny Ramjr Iaarhkejtgs
ζ>
∯ r hζ >
J 0: / >
ζ:
YJKEX @J^[‛Y JELGMY IAMMGLG AC RJ@RJELJ IAMMGLG AC GELKEGG^KEL IKSKM GELKEGG^KEL HGRJ^X@GEX
R^ANMG@ YGXY
R^ANMG@ :
R^ANMG@ >
(
)
Ketglrjtg4 hx 9
J.
9
+ I >
+ I
∯ 9 8 xhx+ >
8 9
(
me 9 x + >
I.
)+ I
: me ( 9 x + > ) + I 9
H. > me ( 9 x + > ) + I
>
Gvjmujtg tfg ketglrjm ac g x +: > x hx . x
>
+:
g + I J. me > > x N. g + I Gvjmujtg
x
I. g
Gvjmujtg
>
+:
+ I
x
H. > x g
+ I
± ∯ me x hx ∔ ± ± ± ± x
J. jrisgi ± ± > ± ± N. 9
R^ANMG@ 7
J. 8 me ( 9 x + > ) + I
R^ANMG@ 9
8
I.
8
I. me ∔ ± ± ± H. jriske ± ±
∯ jritje x hx .
> J. jritje x ∞ me ∔ : + x + I > N. jritje x + > me ( : + x )+ I
I. x jritje x ∞me ∔ :+ x
>
(
+ I
>
H. x jritje x ∞> me :+ x
) +I
YJKEX @J^[‛Y JELGMY IAMMGLG AC RJ@RJELJ IAMMGLG AC GELKEGG^KEL IKSKM GELKEGG^KEL HGRJ^X@GEX Gvjm Gv jmuj ujtg tg tfg tfg ketg ketglr lrjm jm R^ANMG@ 3 2 ± ς / 8. J. N.
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2.>76
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9
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Gvjmujtg
cra` cra`
∯∯ ( x + y ) hx hy. >
:
J. N.
:
R^ANMG@ 6 Gvjmujtg
>
2
z y
∯∯∯ hxhy hz. 2
J. N.
2
2
Ckeh tfg jrgj ac tfg rglkae nauehgh ny tfg iurvgs
y 0
:> x >
x + 8
, tfg x ∞jxks , x 0:, jeh x 0 8.
J. 8 me 3 N. me>8
R^ANMG@ :2
mk`k mk`kts ts
> > y
R^ANMG@ <
R^ANMG@ ;
ac x ias> x hx wk wktf tf
I. 3 me:7 me:7 H. 3 me 8
Ckeh tfg jrgj ac tfg rglkae nauehgh ny aeg maap ac tfg iurvg x >0 y 8 ( :∞ y >). J. ς sq. ue uekt ktss N. ( ς / > ) sq.uekts
I.
( ς / 8 ) sq.uekts
H.
( ς / 6 ) sq.uekts
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