BSBOBPBZBUBQBTBSBOBRBZBUBQBYBXBWBVBN COC1BRCSA6AHAYC2 B4APBRCSA6BGB2 B4BEBDBL B8B1B2BJBBB6 BIA0 BAB3BMBF 1. BAAD I(y)= integraltext b a f(x, y)dx. (1)I(y)? [c, d]CQC6DJARAGB3D5BPCW f(x, y)? [a, b;c, d]CQC6DJA1 (2)I(y)?[c, d]CQC0D7CHI prime (y)= integraltext b a f y (x, y)dxARAGB3D5BPCWf(x, y),f y (x, y) ? [a, b;c, d]CQC6DJA1 2. BAAD F(y)= integraltext b(y) a(y) f(x, y)dx. (1)F(y) ? [c, d] CQC6DJARAGB3D5BPCW f(x, y) ? [a, b;c, d] CQC6DJA4 a(y),b(y)? [c, d]CQC6DJCHa lessorequalslant a(y) lessorequalslant b, a lessorequalslant b(y) lessorequalslant b. (2)F(y) ? [c, d] CQC0D7CH F prime (y)= integraltext b(y) a(y) f y (x, y)dx + f[b(y),y]b prime (y) ? f[a(y),y]a prime (y) ARAGB3D5BPCW f(x, y),f y (x, y) ? [a, b;c, d] CQC6DJA4D6CT? [c, d]CQ b prime (y),a prime (y)AL?CH a lessorequalslant a(y) lessorequalslant b, a lessorequalslant b(y) lessorequalslant b. 3. BAADBKB3C0BQBHA1 integraltext d c dy integraltext b a f(x, y)dx = integraltext b a dx integraltext d c f(x, y)dyARAGB3D5BPCWf(x, y)?[a, b;c, d] CQC6DJA1 B5A0 B0BKBCB9BHBG 1. F(y)= integraltext y 2 y e ?x 2 y dx, CJ F prime (y). BUA5A8D8 e ?x 2 y ?R×RCQC6DJA4D2A5 F prime (y)= integraltext y 2 y (?x 2 )e ?x 2 y dx + e ?y 5 ·2y ? e ?y 3 ·1 =2ye ?y 5 ? e ?y 3 ? integraltext y 2 y x 2 e ?x 2 y dx. square 2. F(y)= integraltext y 0 (x + y)f(x)dx, CDAS f(x) CWC0D7ARA4CJ F primeprime (y). 1 BUA5A8D8 (x + y)f(x)?R×RCQC6DJA4D2A5 F prime (y)= integraltext y 0 f(x)dx +(y + y)f(y) = integraltext y 0 f(x)dx +2yf(y). F primeprime (y)= integraltext y 0 0dx + f(y)+2f(y)+2yf prime (y) =3f(y)+2yf(y). square 3. F(y)= integraltext 1 0 ln radicalbig x 2 + y 2 dx,AOBSBND1BKB3CJF(y),F prime (0)A4A9AUC34BN D1 F prime (0),ABA1AMBTBCALCMDFA1 BUA5AN y negationslash=0CTA4 F(y)=xln radicalbig x 2 + y 2 | x=1 x=0 ? integraltext 1 0 x 2 x 2 +y 2 dx =ln radicalbig 1+y 2 ? integraltext 1 0 (1? 1 x 2 +y 2 )dx =ln radicalbig 1+y 2 ? 1+y integraltext 1 0 ( 1 ( x y ) 2 +1 )d x y =ln radicalbig 1+y 2 ? 1+y arctan 1 y . F(0) = integraltext 1 0 lnxdx = xlnx| 1 0 ? integraltext 1 0 dx =0?lim x→0 xlnx ?1=?1. (∵ lim x→0 xlnx = lim x→0 lnx 1 x = lim x→0 1 x ?1 x 2 =0) D2A5 F(y)= ? ? ? ?1,y=0, ln radicalbig 1+y 2 ?1+y arctan 1 y ,ynegationslash=0. AJC4CFATAPAHA5C5A9AWA0ACAOCZARAUA7A4D9C9C0BND1 F prime + (0) = lim y→+0 F(y)? F(0) y = lim y→+0 [ ln(1 + y 2 ) 2y + arctan 1 y ]= π 2 ; F prime ? (0) = lim y→?0 F(y)? F(0) y = lim y→?0 [ ln(1 + y 2 ) 2y + arctan 1 y ]=? π 2 . B9 F prime (0) ACAL?A1CR f(x, y)=ln(x 2 + y 2 ), AF f y (x, y)= 2y x 2 +y 2 , f BF f y AX AIBZ? [0,1;?∞,+∞]CQACC6DJA4ACC8AVAUC34ARD5BPA1CPBCD0D3B8CVBND1 F prime (0) = integraltext 1 0 f y (x, y)| y=0 dx = integraltext 1 0 dx =0BXBIAHDDAMDBA1BMCUCWAWACAOCZA3AC CBD3A9AUC3 4 ASCJAOB8CVBND1A1A8AI?BND1CEA8DGDCA1AMAUC3 4 ARD5BPA1 square 2 6.(2) integraltext π 0 ln(1? 2αcosx + α 2 )dx (|α| < 1). BUA5CRf(x, α)=ln(1?2α cosx+α 2 ),I(α)= integraltext π 0 ln(1?2α cosx+α 2 )dx. A8D8 1 ? 2αcos x + α 2 greaterorequalslant 1+α 2 ? 2|α| =(1?|α|) 2 > 0, D2A5 f(x, α) ? R× (?1,1) CQC6DJA4CH f α (x, α)= ?2cosx+2α 1?2αcos x+α 2 ?R× (?1,1) CQC6DJ. C5A9 AUC32 C0AQA5 I prime (α)= integraltext π 0 ?2cosx+2α 1?2αcosx+α 2 dx = 1 α integraltext π 0 (1 + α 2 +1 1?2αcosx+α 2 )dx = π α ? 1?α 2 α(1+α 2 ) integraltext π 0 1 1+ ?2α 1+α 2 cosx dx = π α ? 2 α arctan( 1+α 1?α tan x 2 )| x=π x=0 = π α ? 2 α · π 2 =0. ADCW I(α)=CA2?CZA3A4AW I(0) = 0,B9 I(α)=C =0. square 7. C5A9 integraltext b a x y dy = x b ?x a lnx CJ integraltext 1 0 sin(ln 1 x ) x b ?x a lnx dx. BUA5ACB0CR 0 <a<b, A8D8 sin(ln 1 x )x y ? [0,1;a, b] CQC6DJA2AUA6? {0}×[a, b] CQAUA7 sin(ln 1 x )x y D8 0A3A1D2A5 integraltext 1 0 sin(ln 1 x ) x b ?x a lnx dx = integraltext 1 0 sin(ln 1 x )dx integraltext b a x y dy = integraltext b a dy integraltext 1 0 sin(ln 1 x )x y dx AXA9BHx = e ?t C0AQ integraltext 1 0 sin(ln 1 x )x y dx = integraltext +∞ 0 e ?(y+1)t sintdt( AUA5AJCWA4B6BDADBBA7BKB3) = 1 1+(1+y) 2 [?(y +1)sint ? cost]e ?(y+1)t | t=+∞ t=0 = 1 1+(1+y) 2 . ADCW integraltext 1 0 sin(ln 1 x ) x b ?x a lnx dx = integraltext b a 1 1+(1+y) 2 dy = arctan(1 + y)| b a = arctan(1 + b) ?arctan(1 + a) = arctan b?a 1+(1+b)(1+a) . square 3 B4BEAZBL B8B1B7BJBBB6 BIA0 BAB3BMBF A7AGATA2A0BWDACIBBA7BKB3 integraltext +∞ a f(x, y)dxARDFARA1 1. A4AQCXC7DFA1 AF integraltext +∞ a f(x, y)dx BAAD y ∈ [c, d] D8A4AQCXC7ARA4CPBC ?ε>0,?A 0 , AN A prime ,A>A 0 CTA4?y ∈ [c, d],AB| integraltext A prime A f(x, y)dx| <εBJ| integraltext +∞ A f(x, y)dx| <ε. 2. A4AQCXC7ARCCAUA1 CPBCAL? F(x), CUAQ|f(x, y)| lessorequalslant F(x),?a lessorequalslant x<+∞,?c lessorequalslant y lessorequalslant d. CH integraltext +∞ a F(x)dx CXC7A4AF integraltext +∞ a f(x, y)dxBAAD y ∈ [c, d]D8A4AQCXC7ARA1 3. C6DJDFA1 integraltext +∞ a f(x, y)dxCWy ∈ [c, d]CQARC6DJBECZARAGB3D5BPCWf(x, y)?[a,+∞;c, d] CQC6DJCH integraltext +∞ a f(x, y)dxBAAD y ∈ [c, d]D8A4AQCXC7A1 4. BKB3BQBHAJDIA1 integraltext d c dy integraltext +∞ a f(x, y)dx = integraltext +∞ a dx integraltext d c f(x, y)dy ARAGB3D5BPCW f(x, y) ? [a,+∞;c, d]CQC6DJCH integraltext +∞ a f(x, y)dxBAAD y ∈ [c, d]D8A4AQCXC7A1 5. C0AODFA1 I(y)= integraltext +∞ a f(x, y)dx ? [c, d] CQC0AOCH I prime (y)= integraltext +∞ a f y (x, y)dx AR AGB3D5BPCW f(x, y),f y (x, y) ? [a,+∞;c, d] CQC6DJA4 integraltext +∞ a f(x, y)dx AL?CH integraltext +∞ a f y (x, y)dxBAAD y ∈ [c, d]D8A4AQCXC7A1 B5A0 B0BKBCB9BHBG 4.(1). integraltext +∞ 1 x α e ?x dx, a lessorequalslant α lessorequalslant b. BUA5AN a lessorequalslant α lessorequalslant b CH x greaterorequalslant 1CT 0 <x α e ?x <x b e ?x A8D8 lim x→+∞ x b e ?x 1 x 2 = lim x→+∞ x b+2 e x =0 D2A5 x b e ?x = o( 1 x 2 )(x → +∞), AW integraltext +∞ 1 1 x 2 dx CXC7, B9 integraltext +∞ 1 x b e ?x dx CXC7. AKAWBKB3 integraltext +∞ 1 x α e ?x dx ?CKBOa lessorequalslant α lessorequalslant b CQA4AQCXC7. (3) integraltext +∞ ?∞ e ?(x?α) 2 dx (i)a<α<b (ii)?∞ <α<+∞ 4 BUA5(i) AN a<α<bCTA4CLALCZ R, CUAQ?R<a<b<R, ADCWAVADA4 CG a<α<b, AB|α| <R, AKAWAB 0 <e ?(x±α) 2 <e ?(x?R) 2 ,?x greaterorequalslant R,?α ∈ (a, b). A8D8BKB3 integraltext +∞ R e ?(x?R) 2 dxCXC7A4BYA4AQCXC7CCAAAYANA4 integraltext +∞ R e ?(x±α) 2 dxBAAD a<α<bA4AQCXC7A4D2A5 integraltext +∞ ?∞ e ?(x?α) 2 dx = integraltext +∞ 0 e ?(x+α) 2 dx+ integraltext +∞ 0 e ?(x?α) 2 dx BAAD a<α<bA4AQCXC7A1 (ii)AN?∞ <α<+∞,D9C9C0A5AMCA integraltext +∞ 0 e ?(x?α) 2 dx BAAD?∞ <α< +∞B1A4AQCXC7A1AVADCNA6B7AUAR A>0AB lim α→+∞ integraldisplay +∞ A e ?(x?α) 2 dx = lim α→+∞ integraldisplay +∞ A?α e ?t 2 dx = lim α→+∞ integraldisplay +∞ ?∞ e ?t 2 dx = √ π. CL ε 0 = √ π/2,?A>0, CL A 0 = A +1,AACQCYBLDEC0ANAL? α 0 > 0, CU AQ | integraldisplay +∞ A 0 e ?(x?α 0 ) 2 dx ? √ π| <ε 0 ?| integraldisplay +∞ A 0 e ?(x?α 0 ) 2 dx| > √ π 2 = ε 0 AAA4AQCXC7AUA7ARAZDHCYC0AQ integraltext +∞ 0 e ?(x?α) 2 dx BAAD?∞ <α<+∞B1A4AQ CXC7A1BVAW integraltext +∞ ?∞ e ?(x?α) 2 dxBAAD?∞ <α<+∞B1A4AQCXC7A1square (5) integraltext +∞ 0 e ?αx sinxdx (α>0) BUA5AVADCNA6 A>0, AB lim α→0 integraldisplay +∞ A e ?αx sin xdx = lim α→0 e ?αA (?α sinA ? cosA) α 2 +1 = ?cos A D4AAASA4AN A =(2k +1)π(k CWB1B5AKCZ) CT lim α→0 integraldisplay +∞ (2k+1)π e ?αx sin xdx = ?cos(2k +1)π =1. D2A5CL ε 0 = 1 2 > 0,?A>0, CL A 0 =(2[A]+1)π>A,AACQCYBLDEARAUA7C0 ANA4AL? α 0 , CUAQ | integraldisplay +∞ A 0 e ?α 0 x sinxdx ? 1| < 1 2 ?| integraldisplay +∞ A 0 e ?α 0 x sinxdx| > 1 2 = ε 0 AAA4AQCXC7AUA7ARAZDHCYC0AQ integraltext +∞ 0 e ?αx sinxdx BAAD α>0 B1A4AQCXC7A1square 5