Konstruksi Integral Menggunakan Fungsi Sederhana – Pada [A,b]

! " # $ $ % &' (& ) * + , , * - . # , * -/ # " 01 * # , + # +* ,* , $ -/ # " - . # , $ ! 1 1 * * * 1 % . 2 1 1 * * . * $ , $ , 1 " # ) $ , * 2 1 0 1 2 # / # $ , , 1 1 3 4 * 2 $ , *3 4 5 , 1 " # " # # * 2 1 # * " # " ! $ $ % − 0 * # − 3 4 2 1 &$%$ $ " !'() 3 4⊂ ℜ 3 4 − 3 4 2 [ , ], = 1, 2, 3, … , } ≤ ≤ [ , ] ⊂ ( − ( #, + ( ## % < < '< ⋯ < ) = * 0 * 1 2 " − ** " " '!) (+# > 0 .[/, *] − [ , ], = 1, 2, 3, … , } , ', … , )} ≤ ≤ [ , ] ⊂ ( − ( #, + ( ## #+, $ 3 4 ⊂ ℜ # # -3 4 2 1 0 = ( − ( #, + ( ## .[[/, *]} 3 4 6 3 4 7 1 * ( 1− ( 1#, 1+ ( 1## 1, 8 %9 )9 .2 − ( #, + ( #3⋂2 5 − ( 5 #, 5 + ( 5 #3, ≤ ≤ 5 , = 1,2,3, … , − 1 * . [ , ] ⊂ ( − ( #, + ( ##: ** " -'.) − [/, *] [6, 7] ⊆ [/, *] − [6, 7] !

( #+, $ # #/ [/, *] 2 1 * , _', … , ₎ [/, *] ⊂ 9 2) 1− ( 1#, 15 + ( 15 #3 * 6. 2 :− ( :#, :+ ( :#3⋂2 :5 − ( :5 #, :5 + ( :5 #3 : ≤ 6 ≤ :5 1 ≤ ; ≤ 7 . ( <− ( <#, <+ ( <##⋂( <5 − ( <5 #, <5 + ( <5 ## <≤ 7 ≤ <5 ? ? ? %9 )9 .2 − ( #, + ( #3⋂2 5 − ( 5 #, 5 + ( 5 #3, ≤ ≤ 5 = 1,2,3, … , − 1 : = 6 < = 7 * " − pada 3 4 @ = ([ % = /, ], #, … , ([ : , : = 6], :#, ([ : = 6, :5 ], :5 #, … , ([ < , <], <#, ([ < = 7, <5 ], <5 #, … , ([ ) , )= *]#} , 2 : * " . '-) " # − [/, *] [6, 7] @9A − [/, *]9[6, 7] #+, $ # # " 9 [ , ], = 1, 2, 3, … , B} " − [/, *] @ 9 [C1 , C1], 1 D = 1, 2, 3, … , B} " − [6, 7] 6 # # [/, *] [6, 7] 1 2 # # [/, *]9[6, 7] * @9A = [ , ], = 1,2,3, … , B} 9 [C1 , C1], 1 D = 1, 2, 3, … , B} " − [/, *]9[6, 7] : " " % 1 0 2 +%1 $ 3 % " * # * . 1 1 1 * 2 * # . # * 1 1 &$%$ $ " ( '!-) E = F 6 G)_{J HI} # [/, *] ⊂ ℜ K = [ , # $ K₎ = [ ₎ , ₎] _% = ) = E [/, *] % L E 7M N = O 6 ( − # ) J % # # * . * " 4 '!-) E P # – [/, *] ⊆ ℜ . ℜ R DE 7_NM = D R E 7_NM *. L (E + T#7M N = L E 7 M N + L T 7 M N #+, $ E = F 6 G)J HI T = FXUJ 7UGVW R DE 7_NM = F (D6 #( −) # J = D O(6 #( − # ) J = D L E 7M N E + P = F F (F (6:IW 1J X UJ ) J + 7U#GYZ# # * . [\]U1 , ]U1^, U1_D = 1, 2, 3, … , ;U} " − @U = [ , ]⋂[`U , `U] 1 ≤ ≤ , 1 ≤ a ≤ B. ! *R (E + T#7_NM = O O O(6 + 7U#(]1− ]1 # :IW 1J X UJ ) J 9F (F F)_J X_UJ :_1JIW 6 (]₁− ]1 ## + F (F FXUJ )J :1JIW 7U(]1− ]1 ## 9F (6 F F (])_J X_UJ :_1JIW ₁− ]1 ## + F (7XUJ UF F (])J :1JIW 1− ]1 ## 9F (6)_J F _@X_{UJ U}_# + F (7X_{UJ U}F _@)_{J U}_#

C 9F 6 ()_J − # + FX_UJ 7_U(`<sub>U</sub>− `U # 9R E 7_NM + R T 7_NM : " - % 1 0 5 " * b% _{# . # *} 1 1 &$%$ $ " 6 & ' [/, *] → ℜ b% _{[/, *]} d . ℜ + > 0 # [/, *] [ , ], = 1, 2, 3, … , } − [/, *] E = F 6 G)J HI # E d − R E < + 2 1 d b% [/, *] d = (b%_{# L ( #}M N 7 * " 7 . b%([/, *]# b% _{( )} #+, $ (b%_{# R ( #}M N 7 = d (b%_{# R ( #}M N 7 = d' + > 0 # # . [/, *] 6 (b%# R ( #_NM 7 = d # # [/, *] * 2 @ = [ , ], = 1, 2, 3, … , } " − [/, *] # * / E @ [/, *] E d − R E <e_' 2 1 (b%_{# R ( #}M N 7 = d' # # _' [/, *] * 2 @_' = [`<sub>1</sub> , `₁], ₁ D = 1, 2, 3, … , B} * " '− [/, *] # * / _'E_' @_' [/, *] E' d'− R E' <e_' 2 1 # # ( # = min ( ( #, _'( ## D . [/, *] " 9 [ , ], = 1, 2, 3, … , } " − [/, *] E = F 6 G)J HI # * . " [/, *] # E 6 ( # ≤ ( # ( # ≤ '( # @ 2 " − " _'− [/, *] 1 E # * / # * / _' [/, *] * d − R E <e_' d'− R E' <e_' 2 1 d − d' = id − L E + L E − d'i ≤ d − R E + R E − d' <e_'+e_'= + % *+ d = d_': * " 8 * ' [/, *] → ℜ . b%_{([/, *]# '} .b%_{([/, *]#} (b%_{# R D ( #}M N 7 = D(b%# R ( #NM 7 + . b%([/, *]# , ! (b%# R ( + j#_NM 7 = (b%_{# R ( #}M N 7 + (b%# R j( #NM 7 #+, $ + > 0 #ADB b% [/, *] d = (b%_{# R ( # 7 1} + 1 # > 0 * 2 @ = [ , ], = 1, 2, 3, … , } " − [/, *] # * / E @ [/, *] E kd − R E_NM k < _{1 5}e 2 1 E∗= DE # * / @ [/, *] 6 E D ( # − DE( # = D ( # − E( # < D + 1 E∗ *

mDd − L EM ∗ N m = mDd − L DE M N m 9ikDd − D R E_NM ki 9D kd − R E_NM k < _{1 5}1 e < + % *+ b%. 6 R DE_NM = D R E_NM 1 (b%_{# R D}M N = (b%#D RNM + > 0 #ADB b% [/, *] d = (b%_{# R 7 1} + 1 # > 0 * 2 @ = [ , ], = 1, 2, 3, … , } * " − [/, *] # * / E @ [/, *] # ?EE kd − R E_NM k < + 6 ADB b% [/, *] (b%_{# R j( #}M N 7 = d' 1 + 1 # _'> 0 * 2 @' = [`1 , `1], 1 D = 1, 2, 3, … , B} " − [/, *] # * / _'E_' @_' ?EE_' kd_'− R E_NM _'k <e_' # # ∗= B , _'} 2 1 @∗ = @ ⋂@_' " / [/, *] F E∗= E + E_' # * / ∗ @∗ G * # ?EE ?EE_' 1 ( # + j( # − E ( # − E'( # ≤ ( # − E ( # + j( # − E'( # 9+ + + * # H +E' 2 1 k(d + d'# − R E_NM ∗k = k(d + d'# − R (E + E_NM '#k = kd + d'− R E − R E_NM _NM 'k ≤ kd − R E_NM k + kd'− R E_NM 'k <e_'+e_'= + * R( + j#( #7 = d + d' 9(b%# R ( #7 + (b%# R j( #7 : * " !5 . b%([/, *]# . b%_{([*, 6]# . b}%_{([/, *]#} (b%_{# R}n N = (b%# RNM + (b%_{# R}n M #+, $ + > 0 #ADB b% [/, *] d . ℜ + 1 # > 0 * 2 @ = ([ _%, ], # … ([ ₁ , ₁], ₁#} * " − [/, *] %= E = F 6 G)J HI 1 # * / @ [/, *] # E kd − R E_NM k <e_' 6 #ADB b% [*, 6] d'. ℜ + 1 # _'> 0 * 2 @'= ([`%, ` ], # … ([`) , `)], )#} * " _'− [*, 6] `%= `)= 6 E' = F 6 G)UJ oW# * / ' @' [*, 6] E' _d'− R E_Mn '_ <_'e # # ∗₌ 2 1 @∗= @ 9@_'= ([ %, ], # … ([ 1, 15 ], 1#, … , ([ 15) , 15)#} " [/, *# ∪ [*, 6] 1 = * 15 = ` 15 = 9 = # * . E∗ _{= O 6 GH} I 1 J + O 6 GoW ) UJ15 ( # . [/, *# '( # . [*, 6#

'' @∗ _{K = [ , # 9} = K₁ = [ ₁ , ₁# ₁ = * qU = [ U , U# 29 H H / K)= [ ) , )# 15) = 6 E∗ 2 1 * kd + d'− R E_NM ∗k = d + d'− F 6 ( −)J # = _d + d'− F 6 ( −1J #_ − F) 6 ( − # J15 ≤ _d − F 6 ( −1 J #_ + d'− F)J15 6 ( − # 9kd − R E_NM k + _d_'− R E_Mn _'_ <e_'+e_'= + 2 1 (b%_{# R ( #7 = d + d}n _'= N (b%# R +NM (b%_{# R}n M : * " !! 9 $ $ : + 3 ; , . b%([/, *]# ! + > 0 @ = [ , ], = 1, 2, 3, … , } @'= [\`U , `U^, U_a = 1, 2, 3, … , } # − [/, *] E = F 6 G)J HI ! # @ [/, *] E E' = F)UJ 6 GoW $ @_' [/, *] E' kR E −_NM R E_NM 'k < + #+, $ ⟹ + > 0 #ADB b% d . ℜ + 1 # > 0 * 2 * 2 @ = [ , ], = 1, 2, 3, … , D} " − [/, *] # * / E @ [/, *] E kd − R E_NM k <e_' 2 1 2 @_' = [`1 , 1], 1 D = 1, 2, 3, … , } * " − [/, *] # * / E_' @_' [/, *] E' kd'− R E_NM 'k <e_' * kR E −_NM R E_NM _'k = kR E − d + d −_NM R E_NM 'k ≤ mL E − dM N m + md'− L E' M N m <+_{2 +2 = +}+ ⟸ + > 0 # # > 0 [/, *] 2 1 2 @ @_' " − # * E @ # * E @ ′ R E − R E ′ < + 2 1 # # _' 0 < _'< 1 2 @_' @_'′ . " _'− # * / _'E_' @_' # * _'E_' @_'′ R E_'− R E'′ < + 2 1 # # _u 0 < u < '< 1 2 @u @u′ . " _u− # * / _uE_u @_u # * _uE_u @_u′ R E_u− R Eu′ < + " * * v = R E1 D = 1, 2, 3, … } 6 @₁ " − 2 9 = _R E − R EU_ < + * 2 1 0 I 1 1 ) 1 1 ) R Ew $ + 1 < * R E_w− d < + + > 0 # # _w * 2 @_w * " _w− # * . _w @_w Ew

' R Ew− d < + : * " !" A%B ! - [/, *] .%/ b% _{[/, *]} #+, $ + > 0 #ADB J 1 d . ℜ + 1 ∗ ∗_{> 0} * 2 * [/, *] @ = − < ∗ ξ . [ − ] y(z, # − d < + A #B 9F)_J ( #( − # # = ∗> 0 6 ξ . [ − ] ξ . [ − ] ⊂ (ξ − , ξ + # G * J 2 " − 2 1 6 = ( # # * – " 1 > E = O 6 GHI ) J = O ( #GHI ) J ( # − E( # = { ( # − O ( #GHI ) J { 90 < + 1 E *R E_NM = y(z, # 2 1 kR E_NM 7 − dk = y(z, # − d < + : -0 * * *+ " " − # * / 1 1 b% b% 2 * # # . # 1 1 . 3 4 F " A C B 0 1 0 2 3 I , # , 3 4 ) K* J 2 0 A '''B 4 3 2 $5 16 * 7 F " 7 1 3=4 7 J A LB 8 3 ! 2 * , , 1 3L4 F " M;1 1 J A '''B 3 ' 4 9 ! 4 5 16 2 K 1 " K 3&4 A '' B # 2 # : 9 % 7 2 * 1