Билеты матем_ПЕ_1 сем 2011

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n!1

= 1+x

p@o)?3:4/2h)?.12 PsW:4/E?4/y|m1(f x *)?BC2 0A) B GA9 0 Y 2 2

x @

}@o)?3:4/2L9 61E/W ( 68D 68+R x+2619 9 y~3h2 9:4/61m1( ),+

x +4x+3 dx

12+22 ::: n2

@s y ; 2 .8+ 2:4/-wW ( 68D 68+z; 2lim.8+ E/>/E/3LW E/.8+ 68D E/>{)

n!1 n

2 .1W EQ+ -?J1G?IhW ( 68D 68+z2 9:4/6Fm1( ),+f x @-/9 E/3h.FGAPsPsy ` EKD W 2 .1-jW ( 61W EKDx)?>{)Q4/68+*I

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= (cos

i@o)?3:4/2LW ( E/2:JK>/EKD 9:GAy |Dx)?9 9 E/3LB GA9 0 Y 2 2

68+ 619 9 y~3h2 9:4/61m1( ),+

@s` + E/.10A)QI\BC2 mFGA(*) E/m1(*)?9= 4x x + =23@;o)?3:4/2jWA+619*)C0( 2 E?>/y\)Ps2D -C (*)?.1W EQ+ E \619 9 E/3L> >/61(/7:9 613hW EQ+*GAWA+ E/.10AE/.F

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N/@u 61E/( 61P!) EQ+ +*I=@ ` ( 2 Ps61( y 616CW ( 2 Ps619 619 i@o)?3:4/2LW ( E/2:JK>/EKD 9:GAy = arctg|Dx)?9e 9 E/3LB GA9 0 Y 2 2

p@o)?3:4/2L9*)?2 d/EQ+ -? X616C2L9*)?2y =Ps619x x -+?3 X616CJK9*)9*)\E?4/([0; 2]6FJK0:; 6196

}@o)?3:4/2L9 61E/W ( 68D 68+R 619 9 y~3h2 9:4/61m1( ),+

x cos x dx

1 +2 ::: n

@s y ; 2 .8+ 2:4/-wW ( 68D 68+z; 2lim.8+ E/>/E/3LW E/.8+ 68D E/>{)

n!1 n4

2 .1W EQ+ -?J1G?IhW ( 68D 68+z2 9:4/6Fm1( ),+f x @-/9 E/3h.FGAPsPsy ` EKD W 2 .1-jW ( 61W EKDx)?>{)Q4/68+*I

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N/@u% .12 PsW:4/E?4/y|m1(*)?BC2 0A) B GA9 0 Y 2 2h25Ps6F4/E D y

ln ex x

i@o)?3:4/2LW ( 68D lim68+zB GA9 0 Y 2 2

x! sin x

p@o)\W EQ+ E w2:4/68+ -/9 E/3LW OxEQ+*G/E/9*)?3:4/2L4/E/;.12h02:JE/E/(/D0AE?4/E/(0:G2 9*)Q4/9E/3 E E?4/( AB6FJKE/0>/2AD 619LW E Dh9*)?2 d/EQ+.8+(0;22)-;?BX2(0; 8)P: GAm + E/P

}@o)?3:4/2L9 61E/W ( 68D 68+R dx 619 9 y~3h2 9:4/61m1( ),+

x+1

@s y ; 2 .8+ 2:4/E/d/(*)QJKE/>{)?9-wE/d? 1P 4/68+x)9 E/m1Ex >/(y *)Q= X619 2 61P 8+4

>/E/0 (:GAmC61m1EjPs619 -? X613hW EQ+*G/E/.12=@ ` EKD W 2 .1-jW ( 61W EKDx)?>{)Q4/68+*I

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N/@s` 61( >/E/E/d/(*)QJK9*)QI5B GA9 0 Y 2:IL2L9 61E/W ( 68D 68+ 6 ` ( 2 Ps61( y @

i@s` E/0A)QJ2.1W)Q4/EQ+- -?J1G?IhE/W ( 68D 68+;:4/E619lim(22 63 W= (: 68D 68+x)jB G!0

p@se .1.8+ 68D E/>{)Q4/-j9*) PsE/9 E?4/E/9y = x3 9(4E/.x)F4/: -w25 10A.84/( 6

x p

R

}@o)?3:4/2y0(0) 61.8+y2(x) = 1 + t dt :

1

1

@s y ; 2 .8+ 2:4/-wE/W ( R6ln(1+8D x6dx8+ 619 9 y~3h2 9:4/61m1(*),+

1+x

0

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N/@s 4/E/( E/35J )?Ps61;*)Q4/68+c+ 68D .F4/>-/92:Ih>?4/E/(y 3 W ( 68DE/m1E68+ @J )?Ps61;*)Q i@o)?3:4/25GA(*)?>/9 619 2 6C4/E/3L0A)?y = x 0AE?4/E.K)Q4/(*)QI5W*)?(*),+68+-/9 E/3h0L0+(68+2

W ({I yPsE/3= 2x:

p@o)?3p:4/2 >/y .1E?4?G Y 2A+ 2 9AD0AE?4/E/((*) 9*)?2y d/EQ+3PsE,-? w9X61m1EES>/WE/d2

R = 3

}@o)?3:4/2L9 61E/W ( 68D 68+R p 619 9 y~3h2 9:4/61m1( ),+

xdx

2

@se .1W EQ+ -?J1G?ILm161E/Ps6F4/( 2 ; 6K.10>/y2 3; 2.1Psy.R 8+(2 2:4/.jx8+zWj) dx:- ( 68D2

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N/@^61.10 E/9 61; 9 EwP!),+ y 6C2Ld/61.10 E/9 61; 9 Ewd/EQ+ -? X2 i@ # E/0A)QJ )Q4/-jj9sin x61(*)?>/619sin x2 x .xF4/>/E2j; m Dx61; x2 W ( E/2:JK>/EQ+ -/9 y

p@o)?3:4/2L9*)?2 d/EQ+ -? X616C2L9*)?2y =Ps619x x -9*)\E?4/(?X616CJK9*)[ 1;6FJK0A62]: ; 619

}@o)?3:4/2L9 61E/W ( 68D 68+R 619 9 y~3h2 9:4/61m1( ),+

cos x sin x dx:

2

@s y ; 2 .8+ 2:4/-wE/W ( R68D2 e x 68+ex dx:619 9 y~3h2 9:4/61m1(*),+

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N/@c7{61P!)j2 .1.8+ 68D E/>/)?9 2:IhB GA9 0 Y 2 2h2LW E/.F4/( E/6F

cos x 1

i@o)?3:4/2LW ( 68D lim68+zB GA9 0 Y 2 2

x! cos x 1

p@o)?3:4/2L.F4/E/( E/9 y W ({I PsE?GAm1EQ+0AE?4/E/(-/9 2 0A)j9*)?2y3LP!E w9d/EQ+E - >/W 2 .K)Q4/-4j>x y 8+= 4+: 2 W .

}@o)?3:4/2L9 61E/W ( 68D 68+R 3+x 619dx 9 y~3h2 9:4/61m1( ),+

1+x

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E/m1(*)?9 2 ; 619 9= E/3h0x x =( 2: >/y P!2

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N/@s` (*)?>/2A+ E\ E/W 2:4Q),+*I=@ ` ( 2 Ps61( y W ( 2 Ps619 61 i@o)?3:4/2LWp( 68D 68+zW E/J.),Dx)?98+68D E/>{)Q4/68+9E/3L( 610:GA(-/9 E/(.F4/2619:4{9 E '

x1 = 1; xn+1 = 2 + xn ; n 2 N:

p

p@o)?3:4/2LW ( E/2:JK>/EKD 9:GAy = arcsin|Dx)?9x: 9 E/3LB GA9 0 Y 2 2

n!1

2 .1W EQ+ -?J1G?IhW ( 68D 68+z2 9:4/6Fm1( ),+f x @-/9 E/3h.FGAPsPsy ` EKD W 2 .1-jW ( 61W EKDx)?>{)Q4/68+*I

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@s y ; 2 .8+ 2:4/-w9 61.1E/d/.F4/>/619 9 y~3h2 9:4/61m1(*),+¢2A+R dx

(1+x

0

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p@o)?3:4/2LW ( E/2:JK>/EKD 9:GAy = ln(cos|Dx)?9 9 E/3LB GA9 0 Y 2 2

}@o)?3:4/2L9 61E/W ( 68D 68+R 619x 9 y~3h2 9:4/61m1( ),+

x 2) e dx:

=2

@s y ; 2 .8+ 2:4/-wE/W ( 68DR (cos68+x 619sin x 9dx:y~3h2 9:4/61m1(*),+

=2

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p@o)?3:4/2tp.F4/E/( E/9 ykW ({I PsE?GAm1EQ+>/W-/92 2.K)?90A) 9 *)?2E/m1Ez>hWd/EQ+ (*),D R2:G/= .5K)@

}@o)?3:4/2L9 61E/W ( 68D 68+R x+1)619dx 9 y~3h2 9:4/61m1( ),+

x

1

@s y ; 2 .8+ 2:4/-wE/W ( R68Ddx 68+ 619 9 y~3h2 9:4/61m1(*),+

ex e x

0

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p@se .1W EQ+ -?J1G?ILPs9p E/m1E/;A+ 619 y 613A+>/y;E/(*)\W2.8+ 2:4/61(- >/E/m W ( 2 dQ+ 2: \619 9104y ]6CJKY 6F9*)2:4/; 619-wW2:IE/m1( 6F X9 E/.F4{2hW ( 2 d

}@o)?3:4/2L9 61E/W ( 68D 68+R px dx619 9 y~3h2 9:4/61m1( ),+

x

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` ( 2 Ps61( y @

x arctg x

i@o)?3:4/2LW ( 68D lim68+zB GA9 0 Y 2 2

x! x

2 p

@s y ; 2 .8+ 2:4/- E/W ( 68D.1W68+EQ+619-?J1G?Ij61m1ECm161E/Ps6F4/(9 y~3j2 9:4/61m1(*),+R x dx: 20

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i@se .1.8+ 68D E/>{)Q4/f x - B GA9 0 Y 2

( = x

x 3 9*)~>?E?JK(*)?.F4?)?9 2 6 2 GAd/y >{)?9 2 p@o)?3:4/2LW ( E/2:JK>/EKD 9:GAy = e |Dx)?9tg x: 9 E/3LB GA9 0 Y 2 2

}@o)?3:4/2L9 61E/W ( 68D 68+R 1+x 619 9 y~3h2 9:4/61m1( ),+

px dx:

1

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x! 1

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