Toan pt.de075.2012
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1. The document discusses integration and properties of integrals. It shows that the integral of the derivative of a function equals the function evaluated from negative infinity to positive infinity. 2. Several integral properties are demonstrated, including properties related to adding or subtracting integrals and integrating with respect to different variables. 3. The document also explores integrals of functions over all real numbers and shows some integrals equal zero while others do not, depending on the properties of the functions.
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Toan pt.de075.2012
- 1. ! " # = − + !"# $ % &' ( )* &+ ( , ( - !. / # 0 % 1 2 3 4 5 6 76 # ! ( ! 8# 9 : ; ) &+ ( )< &+ ( => 7 $% & ? % &' ( )* 5 5 ! ! ! − − + = ? % &' ( )* 55 5 5 + − + − = − + + − ∈ $% @> >! % 1 A A + + = +∫ $% ' AB ( )C D ( 37 3E7E E !F 37 A # ( ! !1 G 4 &+ ( , ( 7 E / H I % , ( 377E3E (F! J KJ 37 L 33E L # ?8 M6 N6 O AP A&Q A )R ( 0 !"# 77E6 E6 7 S A 9 0 ) !/ 37 # ! 7S L 5 @> T # 0 >! = AB ( )C 37 3E7E E ! D ( ); ( M3 NOS⊥ $% ' # ! -&' ( U# V L D ( ); ( 5 + + ≥ + + + ( ) !" # $ % $ % & "' ( $% ' &* @) ( I % , ( / 9 ) * 6 ! * * :37 !F - 2 >! ; ( 5 7 3 W 4 ( # 0 X !"# # &+ ( ! Y ; ) &+ ( , ( L W @* 8# 9 G 7 @) ( = Z ( ( # H 2 8# 9 : [6 ! I % , ( O J* +− + − = ! ! &+ ( , ( * +− − = = − 6 5 * +− + = = @* ! ! 0 3 ) 7 ) # ! 37 ( ( H I % , ( O 37 ! ! O 9 = ( ; ( $% ' &* @* / ( = Z ( ! D# ) ( = # ) 0 , + = + 6 A (R -&' ( U# V + = ! " $% ' & + & @) ( I % , ( 8# 9 ) * ! & - . / &+ ( , ( 0 1 * 1 / 2 ]# 0 ( 3 - 9 ( ) # ! / (3 ARZ !F 9 - ; ( @* ( 3 # ! ! R # ( ! &(3 A U ^ @) ( = Z ( ( # H 2 / 9 : [6 ! * RZ ( 37 _ H 3 4 4 J 4 4 4 W5 G 7 ; ) I % , ( S J=−++ +* @* / 9 !"# G _ $% ' &+ @* (R -&' ( U# J ` ` 3 3 3 3 + + + + + = + J& = &&&&&&&&&&&&&&& ,- &&&&&&&&&&&&&&&& 4 5 # 6 7 # 89: 3; " 4 5 # # < #= > : ?@ > AAAAAAAAAAAAA:BC ;" AAAAAAA Thi thử Đại học www.toanpt.net
- 2. = − = = ⇔ = − ! ( )−∞ − " ( )+∞ #$ ! ( )− % & $ $ '( & ) *+ ,- . % & $ $ & & ) /*+ ,- ,0/*1 2 3 4 & →+∞ = +∞ + →−∞ = −∞ 5! ) /∞∞∞∞ /* * %∞∞∞∞ ,6 % . / . % , 2 /∞∞∞∞ . %∞∞∞∞ 7 # % 3 "4 8$ 9, 0.+ *1 % 3 "4 8$ 9) 0*+ .1 0/:+ .1 ; <= > <? @ A B( ' $C D C$ E ( )!= − + F ' $G $G <? @ A " # 0-1 E D $G H <= > ( ) ( )( )" ! "! ! ! ! !− + = − + ⇔ − + + − = ! ! ! − = ⇔ + + − = <? @ A $I # 0-1 & ' H J D > H <= > $C D H J D K$ : # # > > ⇔ ⇔ − ≠ ≠ L C M F 50)*+,*1+ -0):+,: + = − = − = − + = − + ' 9 N <? O <? P 5- " $ Q $ = " # %− + − = ⇔ = &'() * +, - . / !0 $ $ 1. π≠ ∈» +2 1 &34 '56 7 &38 ' 9* :'; &56 7 956 7
- 3. <2 . + = − − ⇔ − = − − − = − − ⇔ = − ⇔ − − = :;= :;= = = π π π = ⇔ = ± + ∈ > :' ? <2 9 @ A &) 95B: 7' A :C) '56 7 &38 ' DE π π= ± + ∈ » 0 9 @ A ≤ ≤ F& = ≥ ⇒ + = = − ≥ 0 '56 7 &38 ' &3G &'E ' " " + + = + + ⇔ − + + − = 8$$$$$ = ⇔ + = + = 0 +2 ) / H = − ⇔ = 0 +2 )0H/ &) :I 'A + = + = + = =+ = ⇔ ⇔ ⇔= + = + =+ − − = = = = ⇔ = +, '56 7 &38 ' :I " 7' A !/ !/ = $ D " D D D D D D + + + + + + = = = + + + +∫ ∫ ∫ ∫ = = = −∫
- 4. " D D + = +∫ D D= + ⇒ = + D D + + = = = +∫ D= + = + + − + J DE &3 7 9 KLK &'8 M M M M M M M M ⊥ ⇒ ⊥ ⊥ = 3) 7I: 7 N) L K <E LLK K :'O ' DE 7I: M $ P 3) M Q= $ M $&) M= = I M M M $ M M$ M - -= = " $ M M M M M M $$ $M$ M $ M M= = = 9<&& 0 R R M R R M R R M ⇒ 0 M M M # L = − − = + = ⇒ ⊥ $ F& ' : &'S; :'T 7 ' &34 K ⊥ 4 ⊥ M P 3) ⊥ M U <E = 3) V⊥ + $W; )H 0 H: 0 :) / " 4 + XXX / " " " + + + + + + + + + + + / " " " + + + + + + + +
- 5. 'S; L Y = &) :I " " Z Z + + + + ≥ + + " Z − − ⇒ ≥ + + ;E &;E &56 7 &[ &) :'T 7 ' 95B:. " Z − − ≥ + + " Z − − ≥ + + " 7 <? <2 <? :C) " &) 95B: XXX + + ≥ F& ' : &) -] -E 7 :'T 7 ' 95B: . ) 0 H 0 : ^ " + + / "$ _ 7 &'T: !` 3) ' ) / H / : / : + ) ) :I '56 7 &38 ' 95a 7 &'_ 7 b . !0 / $ 7J ' DE ';` 7 : :' &U &2 L$ = = ⇒ = $ $⇔ = ⇔ = J &;c 9 - DE ( ) &) :I 'A " " + = = = + = ⇔ ⇔ − − = − = = = − W; DE &3 7 9 bL <E 4 0 +2 = 3) L <E " d 0 +2 " − " − = 3) Z " " − − <E " " − +, :I ') :F L &';` * L <E " d ';F: Z " " − − <E " " − ) :I " . " . = + = + = + = = − = − + <S:&6 ' & ? :C) DE ! − "∈ ⇒ + + − "∈ ⇒ + − " "= − + − − + − $ "! = ⇔ Q + + = ( )R " R R " " + − − − − + ⇒ = = = = − ⇔ =
- 6. +2 Z # % " " " − = − ⇒ = ⇒ − − % " " " " − − − = ⇒ = ⇒ − $ +, eeee + ) . ∈ ^ "$ + = ⇔ + = − − 0 ⇔ f f 0 f / Q ⇔ f f / ⇔ = = − +, / $ 0 +2 / &) :I " " " − + = = + = ∑ 0 'S; HE 3) &) :I d 0" / / $ +, => 'c 7 'Y 7 :'T) ! &3; 7 ') &3 DE " #= + H$ J # DE 95a 7 &'_ 7 g ) <E RR $$%&$ # :I &. ' $($) $*$+$%$,- J ./ $0$ 1 &34 $ # =); :'; .$%$2$ ) :I. ( ) ( )Q K f f $ " " " " . . . . .+ ⇒ = − − = ⇒ ) :I :' < &) 7 : $ 34$ $*$ $*$ $%$ $*$ $*$2 /h :' < &) 7 : '( 'i& ' $*$ '( 'i&$ jk& &T 7 : .- W] :I . DE '8 ' H8 ' 'E ' %&$ $*$ $%$ $*$ . J # 9> !T 7 <2 g ) $%&$ $%$ # $%&$ $*$ .$%$ # $*$ .- $*$ '( 'i& ' <E :'l ' # $*$ . '( 'i& ') " 9 #5$ 5$. &'_ 7 'E 7$ W; # 9> !T 7 <2 g ) 4 &8 95B: #. ) $*$' $($66 / &) :I$ ( )M K # f $7 7= ∩ ⇒ − ⇒ $ /h KW. + $*$ $($6'$%$,$ &) :I " M.= ∩ ⇒ $ W; % ".= ⇒ − $ W] &'i ' K W &'_ 7 'E 7 &'8 K L WK :m 7 &'_ 7 'E 7 $ +, ') 9 :n &8 DE. % "− <E " $ 0 J DE &3 7 9 :C) /h " − J L ) H : /h "− − − − + ) :I "∈ ⇒ + + − =
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