36 N G 04 犱犫犮 X, 犐 (, β ) () 犫 ) ( 犮犱 ) β * 6 B, 犐 (, β ) 犮 β 犐 (, β )], ( 犮 ) β ( 犫 ) β ( 犮犱 ) ( 犮 ) β 犐 (, β ), *,,, X, 犐 (, 犮 β ) β ( 犮犱 ) β (

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1 7 04! " # STUDIESINCOLLEGE MATHEMATICS Vol7,No Jan,04 DOI:03969/jissn bh# ' (!!!! /! :, 00875) *! E_,% >00 E %743 E, 0 6 ;! E; FGHI: O7 JKLMN A JO': (04) 犃犖犲狑犐犱犲犳犻犻狋犲犐狋犲犵狉犜犫犲 CAIJunliang (SchoolofMathematicalSciences,BeijingNormalUniversity,Beijing00875,PRC) 犃犫狊狋狉犮狋 : Anewindefiniteintegraltableispresented,whichincludesaltraditionalindefinite integralrecursion formulasforalofthetables,and combines nearly 00 commonly used indefiniteintegralformulasinto43formulas 犓犲狔狑狅狉犱狊 : indefiniteintegral,recursiveformula,integraltable R L :! " ) * [ 0], Q 6, N X & \?:6 7,T 09 ;CDAB: :3*" P(3733);* b (05560GK) E : (957),, `,!,,> " I ( caijunliang@bnueducn E7: ER *, ; 6 7, E/, 7: E "4,0 H8J, RMI (: ; * &S ;, 6 7 E R * 5 E8" 4,%!,6 W! E5 X8"M! E,X L *,7? E 8 N,% \ ; 7:! E_,% >00 E % 43 ES 5 X % $! E O,!Y H : R W!") * TW@,,Rc9 6 ^,c *, 犫, 犮, 犱,,, β, μ 瓗, 犿,, 犻, 犼,, 狉, 狊瓔 * 瓗 7!O, 瓔 73*!O 瓔 e 瓔 o : 瓔 ( 瓔 * _ [!!:!%! μ :3*! 犻, μ 0 μ 0, μ μ μ, μ 犻 μ ( μ) ( μ 犻 ), μ 犻 μ ( μ) ( μ 犻 ), * μ 犻 () 犻 μ 犻 *4!7 C,C! 犫犮 (E Δ 犫 4 犮 R *,!T 73T, L, E!( ( 犫 ) : ( 犫 ), * 0 )! 8! /!! %!! ^ 7:! $%( 犫 ) ( 0)RbH c 6 E7 7,, 犐 (, β ) ( 犫 ) ( 犮犱 ) β d, 犱犫犮 X, ( 犫 ) ( 犮犱 ) β 犐 (, β ) (β) ( β ), 犮 β 狘犫狘 (β)

2 36 N G 04 犱犫犮 X, 犐 (, β ) () 犫 ) ( 犮犱 ) β * 6 B, 犐 (, β ) 犮 β 犐 (, β )], ( 犮 ) β ( 犫 ) β ( 犮犱 ) ( 犮 ) β 犐 (, β ), *,,, X, 犐 (, 犮 β ) β ( 犮犱 ) β ( 犫 ) 犮 β 犐 (, ( β ) )! () ( 犮犱 ) 犿 ( 犫 ) d 犿 0 犮犿 C 犿 ( 犱犮 ) 犿 ( 犫 ) ( ), ( 0) 狘犫狘 ( ) () ( 犫 )( 犮犱 ) 犿 d 犿犿 ( 犱犮 ) 犿 ( 犮犱 ) 犿犫犿 ( 犱犮 ) 犮犱 (3) ( 犫 ) ( 犮犱 ) 犿 d 犿犻犻 () 犻犫犻 ( 犮, 犿 ) 犻犻 ( 犫 ) 犫犻 (, ) 犻犻 ( 犮犱 ) 0(, ) 犫, 犮犱 *, 犫犻 ( μ, ) C 犿犻犿 () 犮犻犿犻 ( 犱犮 ) μ (4) d 槡犫 ( 犮犱 ) 槡犫 ( ) ( 犱犮 )( 犮犱 ) *, ( 3)!! 犐 0 ( )!!( 犱犮 ), 犐 0 d 槡犫 ( 犮犱 ) 狘狘槡狘犮 ( 犱犮 ) 狘 arctan 犮 ( 犫 ) ( 犱犮 > 犫 ), 犱犫犮槡犮 ( 犫 ) 犱犫犮 ( 犱犮 > 犫 ) 槡狘犮犱狘槡 (5) 槡犫 d ( 犮犱 ) ( 5)!( )! 槡犫犮 ( )!( 3)!( 犱犮 )( 犮犱 ) 槡犫犮 ( )( 犮犱 ) ( 3)!! 犐 0 犮 ( )!!( 3)( 犱犮 ), * X 槡犫 d 犮犱槡犫 ( 犱犮 ) 犐 0 犮犮 (6) 槡 ( 犫 )( 犮犱 ) 犐 ( 槡狘犮狘 ( 犫 ) 槡狘狘 ( 犮犱 )) ( 犮 >0), 槡狘犮狘犮 ( arcsin 犫 ) ( 犱犮 < 犫, 犮 <0) 槡狘犮狘槡犫犮犱 (7) 犮犱槡犫 ( 犫 )( 犱槡犮犱犮 ) 犐狘狘 $%( 犫 ) ( 0)& β RbH c 犑 (, ) 6 E7 犑 (, β ) ( 犫 ) β d () ( 犫 ) ( )!! ( ) 犑 (,), *, 犑 (,) 犑 (,0) 犑 (,0), *,,, c 犑 (, ) ( ) ( 犫 ) ( ) 犑 (,0) ( )! (8) ( 犫 ) d 0 () ( 犫 )

3 7 ' :4 P 37 (9) ( 犫 ) 犿 d 0 (0) ( 犫 ) 犿犫 C犿 ( ) (4 犫 )( 犫 ) *, ( 3)!! ( )!! 犫, 犫 arctan ( 犫 >0), 槡犫槡犫犫槡犫 ( 犫 <0) 槡犫槡犫 () 犐槡犫狘槡槡犫狘 ( >0), 槡 arcsin 槡 ( 犫 >0, <0) 槡犫 () 槡犫 d 犫槡犫犐 (3) 槡 ( 3 犫 ) 犫槡犫 (4) 槡 ( 犫 ) 3 d 8 ( 5 犫 ) 槡犫 3 犫犐 8 3 $% 犫犮 & 槡犫犮 RbH Rc E*, W Δ 0 (5) ( 犫犮 ) ( 3)!!( )!(4 ) (Δ) ( )!!( )! 犫 (4 ) ( )!! 犐 3 ( 犫 ) (Δ) ( )( )!!, *, 犐 3 d 犫犮槡 Δ arctan 犫 (Δ <0), 槡 Δ 槡 Δ 狘犫槡 Δ 狘 (Δ >0) 槡 ( 犫 ) Δ (6) d 犫犮狘犫犫犮狘犐 3 (7) 犐 4 槡犫犮狘犫槡 ( 犫犮 ) 狘 ( >0), 槡 arcsin 犫 ( <0) 槡槡 Δ (8) 槡犫犮 d 犫 4 (9) d 槡犫犮槡犫犮 Δ 犐 4 8 槡狘狘犫槡犫犮犐 4 狘狘 4 $% RbH R ^ E*, W 犿狉, 狊, 0 狉, 狊 (0) sin d 犻 () cos d 犻 () csc d!( 犻 )!!cos (!!) ( 犻 )!! sin 犻! ( (!!) 瓔 e),! cos ( (!!) 瓔 o)!( 犻 )!!sin (!!) ( 犻 )!! cos 犻! ( (!!) 瓔 e),! sin ( (!!) 瓔 o) ( )!( 犻 )!!cos 犻 csc 犻 )!!( 犻 )!!] ( )! ( 瓔)!!] e ), ( )! tan )!!] ( 瓔 o) (3) sec d ( )!( 犻 )!!sin 犻 sec 犻 )!!( 犻 )!!]

4 38 N G 04 ( )! ( 瓔)!!] e ), ( )! )!!] π ) ( 瓔 o) (4) sin 犿 cos d 犻犼 ( 犿 )!!( 犿犻 )!! ( 犿 )!!( 犿犻 )!! sin 犿犻 cos )!] ( 犿 )!( 犼 )! ( 犿 )!( 犼 )! cos 犼 sin )!] ( 犿 )! ( 犿 )! ( 犿, 瓔 e), { sin ( 犿瓔 e, 瓔 o),!( 犿 )!! ( 犿 )!! cos ( 犿瓔 o) (5) 犿 sin cos 犻犿犻 ( 犿犻 )!( 犻 3)!!csc 犻 ( 犿 )!!( )!!( 犿犻 )!!cos ( 犿 )! 犿 )!!] 犼犼 ( )!! ( 犿 )!!!! ( 犿犼 )!sin 犿犼 )!!] 犿犼 cos ( 犿 )! 犿 )!!] ( 犿, 瓔 e), π 4 ) ( 犿瓔 e, 瓔 o), 犿犼 sec 犿犼狘 sin 狘 ( 犿, 瓔 o), tan ( 犿瓔 o, 瓔 e) (6) 犿 < X, sin 犿 d cos 犻犻犿犻 () ( 犿 )!!( 犻 )!sin ( )!!( 犻 )!!( 犿犻 )!!cos ( () 犿 )!!( )! ( )!!( )!! ( 犿 )cos { 犼犿 ( 犿瓔 o), 犻 ( 犿 )!( 犿犼 )! 犿 )!!( 犿犼 )!!] 犿犼 } sin ( 犿 )! cos 犿 )!!] ( 犿, 瓔 e), π 4 ) ( 犿瓔 e, 瓔 o) (7) 犿 X, sin 犿 d cos 犻犻犼犻犿犻 () ( 犿 )!!( 犻 )!sin ( )!!( 犻 )!!( 犿犻 )!!cos ( () 犿 )!!( )! ( )!!( )!! 犻 ( 犿 )!!( 犿犻 )!!cos ( 犿 )!!( 犿犻 )!! sin 犿犻 ( 犿 )! ( 犿, 瓔 e), 犿 )!!] { cos ( 瓔 e, 犿瓔 o), 犼 () 犼 C cos 犼犼犻狘 cos 狘 ( 犿, 瓔 o), 犿犻 sin 犿犻 π 4 ) ( 犿瓔 o, 瓔 e) (8) sin sin 犫 d sin( 犫 ) ( 犫 ) sin ( 犫 ) ( 犫 ) ( 狘狘狘犫狘 ) (9) sin cos 犫 d cos( 犫 ) ( 犫 ) cos ( 犫 ) ( 犫 ) ( 狘狘狘犫狘 ) (30) cos cos 犫 d sin( 犫 ) ( 犫 ) sin ( 犫 ) ( 犫 ) ( 狘狘狘犫狘 ) (3) 犫 sin tan 槡 arctan 犫 ( 狘狘 > 狘犫狘 ), 槡 tan 槡犫槡 ( 狘狘 < 狘犫狘 ) tan 犫槡 (3) 犫 cos arctan ( tan 槡槡犫 ) ( > 犫, 狘狘 > 狘犫狘 ), 槡槡犫槡犫 (33) sin 犫 cos tan 槡犫 ( < 犫, 狘狘 < 狘犫狘 ) tan 槡犫槡犫 sin 犫 cos sin cos

5 7 ' :4 P 39 (34) cos 犫 sin ± 犫 arctan ( tan )( 犫 >0), 槡犫槡 ± tan 槡 ( 犫 <0) 槡犫 tan 槡 (35) sin d 犻 0 犻 () 犻! 犻 ( 犻 )! 犻 )sin cos ] (36) cos d 犻 0 犻 () 犻! 犻 ( 犻 )! 犻 )cos sin ] (37) arccos d 犼 0 犼!!( 犼 )! 槡 ( )( )!! 犼 )!!] 犼 ( )!! 狊!! ( )( )!! arccos arcsin d π arccos d (38) arctan d 狊 arctan 犼犼犼 () ( )( 犼 ) 犼狊 () ( )( 瓔 e), arctan ( 瓔 o) arccot d π arctan d 5 $%' U RbH (39) e d (40) d ( )!e! 犻 0 (4) e sin 犫 d 犻 0 犻 0 犻! ( 犻 ) 犻犻 () 犻 ( 犻 )! 犻![ sin 犫 ( 犻 ) 犫 cos 犫 ]sin 犻 e 犻犫 ( 犻 )! 犼 0 犫 [ 犫 ( 犼 ) ] (4) e cos 犫 d 犻 0 犻![ cos 犫 ( 犻 ) 犫 sin 犫 ]cos 犻 e 犻犫 ( 犻 )! (43) 犿 d! 犿犻 0 犼 0 犫 [ 犫 ( 犼 ) ] 犻犻 () 犻 ( 犿 ) ( 犻 )! ( 犿 ), ( 犿 ) _`JK [] 9! " :` [M] : * 3,979: []! " 3:` [M] : * 3,003: [3],, [!" C:` [M] : 9 T" 3,008: [4], Z!"[M] :* b 3, 008: [5] T! " :` [M]Z : $ " 3,985:59 6 [6]! " :` [M] : 9 " 3,99: [7] *!" :` [M]` :" T " 3, 993: 33 [8], Z, %! " :` [M] : 3,985:3 40 [9],!" :` [M] : 3,986:67 74 [0]a/, ", *!" :` [M] : T" 3,989:56 8 []!" :` [M] : T " 3,990:0 3 []#$9 T"!"!" :` [M]3 : 3,007: [3] ` " :` [M] : 9 T " 3,006:85 97 [4], ;! " 3:` [M] : 3,00:63 07 [5],! " [M] : 3,990:50 57 [6]WZT"!"!":` [M]6 : 3,008:36 37 [7],!" B:` [M] : 9 T" 3,009: [8] X,! " ) [M] : " 3,0:8 9 [9] ", :` [M] : 3,000:8 96 [0]?&, *! " :` [M] : " 3,004:33 349

( ) Vol.54 No JournalofXiamenUniversity (NaturalScience) Mar.2015 doi: /j.issn R, ( ]a G,9O ) 56

( ) Vol.54 No JournalofXiamenUniversity (NaturalScience) Mar.2015 doi: /j.issn R, ( ]a G,9O ) 56 154 12 ( ) Vol.54 No.2 2015 3 JournalofXiamenUniversity (NaturalScience) Mar.2015 doi:10.6043/j.issn.0438 0479.2015.02.018 + 9 R, ( ]a G,9O 361005) 56: % 5 ) B` M 7 #

36 N G 04 犱 犫犮 X, 犐 (, β ) () 犫 ) ( 犮 犱 ) β * 6 B, 犐 (, β ) 犮 β 犐 (, β )], ( 犮 ) β ( 犫 ) β ( 犮 犱 ) ( 犮 ) β 犐 (, β ), *,,, X, 犐 (, 犮 β ) β ( 犮 犱 ) β (

36 N G 04 犱犫犮 X, 犐 (, β ) () 犫 ) ( 犮犱 ) β * 6 B, 犐 (, β ) 犮 β 犐 (, β )], ( 犮 ) β ( 犫 ) β ( 犮犱 ) ( 犮 ) β 犐 (, β ), *,,, X, 犐 (, 犮 β ) β ( 犮犱 ) β (