tut3b - i u i • w v ”  w ƒ u x w t g … x w “ ‡ s...

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Unformatted text preview: i u i • w v ”  w ƒ u x w t g … x w “ ‡ s i g  ‰ u ƒ v u ‚ ’ † v u y q ‘ g  ‰ t w ˆ g ‚ s ‡ g g i † t g ‚ s g … „ g ƒ v g p ‚ g  v s € y g x w v u t s r g q p i g h g f e I T d 3( 7 "'9  $ 7!  @ c  " A A I 6 & ' & % " & '   " 2   '  ) " $ @ ! %  " & 2 4 3 3 ( $ " 0 ( ' ( 0  ) 2  " !   " ' %  ( &  )  ! &  " &bS ! $ & $ ' % " ' & 0 " & ' ! %  0 ' % ! g 0 (sinn x)dx = π /2 x = π/2 n→+∞ 0 lim (sinn x)dx = VI 6% 0 π /2 " (  " 3 ( 7  "'% ' ' ( G % a 6%   ) A x ∈ [0, π/2) g (x) = 1 g (x)dx, π /2   " 6 &  7 2 3 ) & ' H   " "! $ Q 7 '% ! $ 4( ! ` D Y Y X W  ) " $ n→+∞ lim g (x) = 0 n→+∞ lim (sinn x) = g (x), UI x ∈ [0, π/2) '  ) ! $ " 2 ' 33 ( & 6 sin x < 1 n → +∞ A  0 '%!  $ " ! % ! %  " 43  1% "% $ "'% " (  "  " & P 0 In = π /2 (sinn x)dx.   ) n→+∞ lim In = 0, 8 7 ' % ) & 3 3 & 6   " ! % T G & 6 2   % 3 5 G % ! (S ' & % " !  $ # 2 ' & 0  !    π 3π π/3 ln x dx = π, ln x + ln(4π − x) xr 2π/3 π xr dx = . + (π − x)r 6 8 43%! (  !3 ( 7  "'% 7 '% ) &33 & 6   "  " ( $3 (1  & ! 3 ( ' ( 0  F 0 π sinr x dx = . sin x + cosr x 4 r π /2 A 1dx = b − a ⇒ I =  @ G $' 3(  4' ( &6 ! $  b b−a . 2 f (x) + f (a + b − x) dx f (x) + f (a + b − x) f (x) dx + f (x) + f (a + b − x) a b A 4 " % 3 ( $ #  " !    "  1 (   )  3 @ ( % (1 4 G G $ 2 ( " ! $ Q ! % b a f (a + b − x) dx f (x) + f (a + b − x) RI = f (x) dx = f (x) + f (a + b − x) f (a + b − y ) − dy = f (y ) + f (y ) a b f (a + b − y ) dy. f (y ) + f (y ) I !  G & 0  @ 3 ( 7  "'% "!    " '    x=a+b−y 8 !3 ( 7  "'% & )"   " 6 & G $!   " )& '  2%! ' & r a = a b = a b I +I y 0 '%! )& P a b ' & % " $ "% " ! @ $ !   "  H ( G  F ED DC B a f (a + b − x) b−a dx = . f (x) + f (a + b − x) 2 b 4'( &6 f (x) dx = f (x) + f (a + b − x) x ∈ (a, b) A b f (x) = −f (a + b − x) '" a I= 2 ' ( ' &% "0 ' $6 3 @ ( 7  "'% ' ( !% f (x) 69 8 7 '% ) &33 & 6   " 6 & ! ( 0 3 ( % 0  5 ! ( 433 ( $ " 0 ( ! % 2   1 & 0  ) " (  " ' &% "!  $ # "!       ¤¦ £  ¢  §¥£  © ¨ § ¦ ¥ ¤ £ ¢¡ =1 n→+∞ lim π 2n In =1. A ¢ ¢ A  1(   ) T G  &    0% ) 2 ' ( S G  &     £   $ #   " 4 @ &! n n+1 "$ V lim n In ≤ lim 1. ≤ lim π n→+∞ n→+∞ n+1 2n A  1(   ) " $ &  7 $ &  " ! "% G%3   " 7 '% H ( A π , 2n "$&7$& " A 7 '% 2 % 1% 2 4 @ " (  " ! %3 5 G% 0 %  )  1(   ) ! ' &% " ( $ #  & )" "! (3   "  '% @ G & 0  ) 69 π ⇒ In−1 ≥ 2n π . 2(n + 1)  1(   ) 43 ( 3% G%! ) & P π . 2n A π ⇒ In ≤ 2n A sin x ≤ 1∀x ∈ [0, π/2] In ≤n−1  1(   ) ! $  " 2 ' (  0 '% ¢ π 2n . π , 2 nIn In−1 = (n − 1)In−1 In−2 . 4(! & " !% "(   limn→+∞ n→+∞ In n ≤ π ≤ 1. n+1 2n π 2n π ≤ In ≤ 2(n + 1) In−1 In−1 ≥ In−1 In = In In ≤ In In−1 = In In−1 = " (  " ! %3 5 G% 0 %  ) nIn In−1 = (n − 1)In−1 In−2 = (n − 2)In−2 In−3 = ... = 1I1 I0 = I 1(   ) 0 '  ¡ "' ( "! ' & 0 !% A 1(   ) ' &% "( $ #    " 6 & !  2%!  " & @ & " In−1 (nIn In−1 )  0 '  $ # !   " 4 @ 7 '% 43 5% "3 $ G ) & P nIn = (n − 1)In−2 . 1(   ) 0 '  ¡ = (n − 1)[In−2 − In ] 0 = (n − 1) π /2 (sinn−2 x)(1 − sin2 x)dx 0 − cos x(n − 1) sinn−2 x cos xdx (sinn−1 x)(sin x)dx I  1(   F ! " ( 5 4 @ ' &% " ( 7  " '% 7 '% ! $ π /2 In In A − 0 π /2 (sinn x)dx = In = &6 (3 $ G &6 0 '  I 6 &  1 $ 0   " 0 "  H !  ) 69 A $ 0  ( 5 $ "  ! " !  ! $ "  3 ! %  "  1 & 5 & n → +∞ &6 '  " = π /2 '   ) ! 1(   @ I  H%3 !  1(   @ ) &  ! $ ! 33  " 0 %  ) In lim n A 1 $0   "  7 (3 = pi/2 [− cos x sinn−1 x]0 0 In π 2n π 2n n→+∞ = 1, In A I V "(  " )& ! '  1  ' ( 0  ) "0 ( 6 '9 43 "' ( 7 3  4  1 2  1& 5  @ ' ( 0  1& @ (   " ! ( 0 4 ' ( '% " $ ...
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