NGUYN HM V TCH PHNBI 1. BI TPSDNGCNG THC NGUYN HM, TCH PHN I.Bng cng thc nguyn hm m rng( )1111ax bax b dx c ,a + + | `+ + + . ,( ) ( )1cos ax bdx sinax ba+ + + c1 dxln ax b cax b a + ++ + c( ) ( )1sinax bdx cos ax b ca+ + +1ax b ax be dx e ca+ + +( ) ( )1tg ax bdx ln cos ax b ca+ + +1ax b ax bm dx m caln m+ + +( ) ( )1cotg ax bdx ln sinax b ca+ + +2 21 dx xarctg ca aa x ++( )( )21 dxcotg ax b casin ax b + ++2 212dx a xln ca a xa x+ +( )( )21 dxtg ax b cacos ax b + ++( )2 22 2dxln x x a cx a + + ++2 2x xarcsin dx x arcsin a x ca a + +2 2dx xarcsin caa x +2 2x xarccos dx x arccos a x ca a +2 21 dx xarccos ca ax x a +( )2 22x x aarctg dx x arctg ln a x ca a + +2 22 21 dx a x aln ca xx x a+ + ++( )2 22x x aarc cotg dx x arc cotg ln a x ca a + + +( ) ( )bln ax b dx x ln ax b x ca| `+ + + + . ,( )12dx ax bln tg csinax b a+ ++2 2 22 22 2x a x a xa x dx arcsin ca + + ( )12dx ax bln tg csinax b a+ ++( )2 2axaxe a sin bx b cos bxe sin bx dx ca b ++( )2 2axaxe a cos bx b sin bxe cos bx dx ca b+ ++1II. NHNG CH KHI S DNG CNG THC KHNG C TRONG SGK 12Cc cng thc c mt trong II. m khng c trong SGK 12 khi s dng phi chng minh libng cch trnh by di dng b . C nhiu cch chng minh b nhng cch n ginnht l chng minh bng cch ly o hm1. V d 1: Chng minh: 2 2dx 1 x aln c2a x ax a ++; 2 2dx 1 a xln c2a a xa x+ +Chng minh: 2 2dx 1 1 1 1 dx dx 1 x adx ln c2a x a x a 2a x a x a 2a x ax a | ` | ` + + + + . , . , ( )2 2dx 1 1 1 1 dx d a x 1 a xdx ln c2a a x a x 2a a x a x 2a a xa x| ` +| ` + + + + . , . , 2. V d 2: Chng minh rng: ( )2 22 2dxln x x ax a + ++ + cChng minh: Ly o hm ta c:( )( )2 22 22 21 x aln x x a cx x a+ + ]+ + + ]+ + 2 22 2 2 2 2 2 2 2 2 21 x 1 x x a 11x x a x a x x a x a x a+ +| ` + + + + + + + +. ,3. V d 3: Chng minh rng:2 2dx 1u caa x ++ (vi xtg ua)t xtg ua , ( )u ,2 2 ( )( )2 22 2d a tg udx 1 1du u ca aa xa 1 tg u +++ 4. V d 4: Chng minh rng: 2 2dxu ca x + (vi xsin ua , a > 0)t xsin ua ,u ,2 2 ] ] ] ( )( )2 22 2dx d a sin udu u ca xa 1 sin u + Bnh lun: Trc nm 2001, SGK12 c cho s dng cng thc nguyn hm 2 2dx 1 xarctg ca aa x ++v 2 2dx xarcsin caa x +(a> 0) nhng sau khng ging bt c nc no trn th gii, h li cm khng cho s dng khi nim hm ngc arctg x, arcsin x. Cch trnh by trn khc phc lnh cm ny.III. CC DNG TCH PHN N GINIII.1. CC K NNG C BN:1. Biu din lu tha dng chnh tc: 21nnx x; m mn n k m mn nkx x ; x x 1nnnn1 1x ; xx x ;mnn m1xx; mnkn k m1xx2. Bin i vi phn: dx d(x 1)d(x 2) d(x p)adxd(ax 1)d(ax 2)d(ax p)( ) ( )x p 1x 1 x 2dx d d da a aat | `t t . ,LIII.2. CC BI TP MU MINH HO1. 3dx1xx ( )321 1 1dx 1 dx1 1xx xx x +| ` + + + . , ( )( )2 3 211 11 dx ln 11 3 2d xx x x x x x cx+ + + + + + + 2.( )14 7 dx = 4 7 7 4 7 dx4x x x x ++ ] + ] ( ) ( ) ( ) ( ) ( )3 5 3 12 2 2 21 1 2 24 7 7 4 7 4 7 4 7 7 4 716 16 5 3x x d x x x c ]] + + + + + + ] ] ]3. ( )( )( )172 2 2d 2d 12 5 22 5xxIxx ++ 1 10arctg510x c| ` + . ,4. ( )( )( )xdx 1 2 1 1 1 1 22 lnln 2 5ln 2 5ln 2 2 + 5 2 2 5 2 52 2 5x xxx x xx xdd c| ` + + + . , + 5.( ) ( )53 2 3coscos 1 sin 1 sin cos cos sin dx1 sinxdx x x dx x x x xx ] + + ] ( ) ( ) ( )3 42 3sin cos1 sin sin cos cos sin3 4x xx d x xd x x c + III.3. CC BI TP DNH CHO BN C T GII( ) ( ) ( ) ( )1x 1 x 2 x 3 x 4J dxx x+ + + + ; 27x 3J dx2x 5+ ; 233x 7x 5J dxx 2 +( )3 2 2 24 5 6102x 5x 7x 10 4x 9x 10 2x 3x 9J dx ; J dx ; J dxx 1 2x 1x 1 + + + ( ) ( )3 2 3 27 815 30x 3x 4x 9 2x 5x 11x 4J dx ; J dxx 2 x 1 + + + + ( ) ( ) ( ) ( ) ( ) ( ) + + + d x 1 x 2 5 x 3 x J ; d x 2 x 5 1 x J ; d x 1 x 3 x J3 321 11 5 21 03 1 0 093( )( )( )( )2432 4 55 912 13 1447x 3x 5J 2x 3 . x 1 dx ; J dx ; J x . 2x 3 dx2x 1 + + ++ ( )9 315 16 174 2 2105x x xJ dx ; J dx ; J dxx x 1 x x 12 3x + ( ) ( )( ) ( ) ( ) ( )18 19 202 2 2 2dx dx dxJ ; J ; Jx 2 x 5x 2 x 6 x 2 x 3 ++ + + ( ) ( ) ( ) ( ) ( ) ( )21 22 232 2 2 2 2 2x dx dx dxJ ; J ; Jx 3 x 7 3x 7 x 2 2x 5 x 3 + + + ln 2 ln 2 ln 2 ln 22x xx24 25 26 27xx x1 0 0 0dx e dx 1 eJ ; J ; J e 1dx ; J dx1 ee 1 e 1 + + + ( ) ( )2 2x x1 1 1 1x28 29 30 31x 2x 2x x 3x0 0 0 01 e dx 1 ee dx dxJ ; J ; J ; J dx1 e 1 e e e e+ + + + + ln 2 ln 4 1 e3x32 33 34 35x 3 x x x0 0 0 1dx dx e dx 1 ln xJ ; J ; J ; J dxx e e 4e 1 e+ + + ( )3 1 165 2 5 3 3 236 37 380 0 0J x 1 x dx ; J x 1 x dx ; J x 1 x dx + ( )2x1 1 1 12x x39 40 41 42x x x x0 0 0 02 1 dxdx dxJ ; J ; J ; J e 1 e dx4 3 4 2 4 + ++ + BI 2. TCH PHN CC HM S C MU S CHA TAM THC BC 2 A. CNG THC S DNG V K NNG BIN I1. ++ 2 2du 1 uarctg ca au a 4. + du2 u cu2. ++ 2 2du 1 u aln c2a u au a5. ( ) + >2 2du uarcsin c a 0aa u3. + + 2 2du 1 a uln c2a a ua u6. + t +t22duln u u p cu pK nng bin i tam thc bc 2:1. ] | `+ + + ] . , ] ]2222b b 4acax bx c a x2a4a 2. ( )+ + t + t22 2ax bx c mx n pB. CC DNG TCH PHN 4I. Dng 1: 2dxA=ax + bx + c1. Phng php:( )+ ++ ++ + 2 22dx dx 1 mx narctg cmp pax bx cmx n p

( )+ ++ ++ ++ 2 22mx n p dx dx 1ln c2mp mx n pax bx cmx n p2. Cc bi tp mu minh ha( )( )( ) ( )12 2 22d d 1 d 2 2 1 2 2 3ln2 4 8 1 4 3 2 2 32 2 32 2 3x x x xA cx x xxx+ + ++ + + ++ + 3. Cc bi tp dnh cho bn c t gii:12dxA3x 4x 2 ;2 32 2dx dxA ; A ;4x 6x 1 5x 8x 6 + + + 2 1 14 5 62 2 21 0 0dx dx dxA ; A ; A7x 4x 3 6 3x 2x 4x 6x 3 + + + II. Dng 2: ( ) 2mx + nB = dxax + bx + c1. Phng php: ( )( )( )+ + + + + + + 2 2m mb2ax b nmx n2a 2aB dx dxax bx c ax bx c ( )+ +| `+ + + . ,22d ax bx cm mbn A2a 2aax bx c | `+ + + . ,2m mbln ax bx c n A2a 2aCch 2:Phng php h s bt nh (s dng khi mu c nghim) Nu mu c nghim kp 0x x tc l 2 20( ) ax bx c ax x + + th ta gi s: ( )+ + + +2 200mx nxx xax bx cx x Quy ng v phi v ng nht h s hai v tm , .Vi , va tm ta c: ( )++ + 2mx nB dxax bx c ln +00x x cx x Numuc2nghimphnbit1 2, x x :21 2( )( ) ax bx c ax x x x + + th tagis+ + + +21 2mx nxx x x xax bx c 5Quy ng v phi v ng nht h s hai v tm , .Vi , va tm ta c: ( )dx++ + 2mx nBax bx c ln ln + +1 2x x x x c 2. Cc bi tp mu minh ha:122x + 3B = dx9x 6x + 1 ( )( )2 2 21 1118 61 18 6 d 11 d9 3d9 3 9 6 1 9 6 1 9 6 1xx x xxx x x x x x + + + + + ( ) ( )( )( )22 21 9 6 1 11 3 1 2 11ln 3 19 9 9 9 3 1 9 6 13 1d x x d xx cx x xx + + + + 3. Cc bi tp dnh cho bn c t gii:( ) ( ) ( )1 2 32 2 27 3x dx 3x 4 dx 2 7x dxB ; B ; B4x 6x 1 2x 7x 9 5x 8x 4 + ;III. Dng 3:2dxC =ax + bx + c1. Phng php: B :ln22duu u k cu k + + ++Bin i nguyn hm v 1 trong 2 dng sau:( )( ) ( )22 2dx dx 1ln C mx n mx n k cmax bx cmx n k + + + + ++ ++ + ( )( )2 22dx dx 1arcsin 0mx nC pm pax bx cp mx n+ >+ + + 2. Cc bi tp mu minh ha:( )( )232 2d 1 d 55 45ln4 162 4454 10 55416x xC x x cx xx + + 3. Cc bi tp dnh cho bn c t gii:1 2 32 2 2dx dx dxC ; C ; C3x 8x 1 7 8x 10x 5 12x 4 2 x + IV. Dng 4:( )2mx + n dxD =ax + bx + c1. Phng php:( )2 22 dxdx2 2ax bm mbDa aax bx c ax bx c+ + + + + ( )222 2d ax bx cm mbCa aax bx c+ + + +2. Cc bi tp mu minh ha: 6 D1 =( ) ( )1 1 12 2 20 0 04 d 2 d d24 5 4 5 4 5x x x x xx x x x x x+ + ++ + + + + + ( )( )( )( )1 1 1 22 22 200 01 d 4 5 d2 4 5 2 ln 2 4 524 52 1x x xx x x x xx xx+ + + + + + + + + ++ ++ + ( ) ( )3 1010 5 2ln 3 10 2ln 2 5 10 5 2ln2 5+ + + + ++

3. Cc bi tp dnh cho bn c t gii:( ) ( ) ( )1 2 32 2 25 4x dx 3x 7 dx 8x 11 dxD ; D ; D3x 2x 1 2x 5x 1 9 6x 4x + + V. Dng 5:( )2dxE =px + q ax + bx + c1. Phng php:t 21 dt 1 1dx ; px q p x qt p tt | `+ . ,. Khi :( )22 2 22dt pt dx dtEpx q ax bx c t t1 a 1 b 1q q ct t p t p t+ + + + +| ` | ` + + . , . , 2. Cc bi tp mu minh ha:( )3122dxE =x - 1 x - 2x + 2. t 22 111 131 ;2dxx ttx tx xt tdtt + ' Khi : ( )( ) ( )1 2 3212 22 1dt t dxE1 x-1 x 2x 2t 1 t 12 2t tt ++ + + ( )11221 21 2dt 1 5 2 2 2ln t t 1 ln 1 2 ln ln21 5t 1+ + + + + ++3. Cc bi tp dnh cho bn c t gii:( ) ( ) ( )2 3 31 2 32 2 21 2 2dx dx dxE ; E ; E2x 3 x 3x 1 3x 4 2x 3x 7 x 1 x 1 + + + + + VI. Dng 6:( )( )2mx + n dxF =px + q ax + bx + c71. Phng php:( )( )( )( )2 2dxdxmqmpx q nmx n p pFpx q ax bx c px q ax bx c| `+ + +. , + + + + + + ( )2 2dx dx mq mq m mF n C n Ep pp pax bx c px q ax bx c| ` | ` + + . , . ,+ + + + + 2. Cc bi tp mu minh ha:( )( )11202 3 d1 2 2x xFx x x+ + + +( )1 12 20 0dx dx2 2I Jx 2x 2 x 1 x 2x 2+ ++ + + + + 120dx2 2Ix x+ +( )( ) ( )1 12020dx 2 5ln 1 1 1 ln1 21 1x xx+ + + + + ++ +( )1201 2 2dxJx x x+ + +. t 20 111112dxx tx txtdtt + ' . Khi :( ) ( )1 2 12 122 21 21 1 2dt t dt 2 2 2J ln t t 1 ln1 51t 11 11 2 1 2t tt + + + ++ + + F1 2I + J( )( )( )2 5 2 2 2 2 9 4 52ln ln ln1 2 1 51 2 1 5+ + ++ + ++ +( )( )( )( )3 2225 12 12 22 1 4 3xdxx x x+ ++ -3 222-2x + 3 dxF =2x + 1 -x - 4x - 3

( )3 2 3 22 22 21 dx 5 dx 1 5I J2 2 2 2x 4x 3 2x 1 x 4x 3 + + + 3 2224 3dxIx x ( )( )3 23 2222dxarcsin x 261 x 2 + +( )3 2222 1 4 3dxJx x x+ . t 21231 13 12 12 222x ttx tx x ;t tdtdxt + '( ) ( )( ) ( )12 1322 213 1213 132 212 12dt 2t dtJ1 5t 6t 11 1 11 2 1 34 t tt1 dt 1 5t 3 1 2 1arcsin arcsin arcsin2 3 45 5 53 2t5 5 + | ` . , + 8 Vy ( ) 255 1 2 1F I J arcsin arcsin2 2 12 2 3 4 + + 3. Cc bi tp dnh cho bn c t gii: ( )( )( )( )( )( )1 1 11 2 32 2 20 0 04x 7 dx 6 7x dx 7 9x dxF ; F ; F8 5x 3x 4x 2 2x 5 x x 4 4x 3 2x x 1+ + + + + + + VII. Dng 7 : ( )2 2xdxG =ax + b cx + d1. Phng php:t 22 2 2 2t d t dtt cx d t cx d x ; x dxc c + + Khi : ( ) ( )2 2 221 1 1 t dt dtG Acc at bc ad cat db tc ] + + ] ] 2. Cc bi tp mu minh ha:( )112 20xdxG =5 - 2x 6x + 1. t 20 16 1 1 76x tt x x tx dx t dt + '. Khi :

( )( )77 712 221 1 13 4 71 t dt 1 dt 1 1 4 t 1G ln ln6 2 2 8 4 t 164 t16 t 5 4 7t3++ | ` | ` . , . , 3. Cc bi tp dnh cho bn c t gii:( ) ( ) ( )2 2 11 2 32 2 2 2 2 21 1 0x dx x dx x dxG ; G ; G4x 3 5 x 5x 11 7 3x 8 7x 2x 1 + VIII. Dng 8:( )2 2dxH =ax + b cx + d1. Phng php:t ( )2 2 2 2 22 22d td.dtxt cx d xt cx d x xdxt ct c + + ( )( )( )22222td.dt t c dx xdx dtx xtt ctd t ccx d +. Khi ta c: ( )( )( )22 222dx dt dtH Aadbt ad bcax b cx db t ct c | ` + + ++ . , 2. Cc bi tp mu minh ha:9( )312 22dxH =x - 2 x + 3. t 2223333722x txxt x txx t + + ' v ( )( )2 2 2 2 2 22 223 3t dtx t x 3 t 1 x 3 x x dxt 1t 1 + ( )( )( )222223t dt t 1 dx x dx dtx xt t 13t t 1x 3 +. Khi ta c:2 3127 2dt2 5Ht ( ) ( )( ) ( )2 37 21 2 5 1 2 2 15 14 2 5ln ln2 10 2 5 2 102 2 15 14 2 5tt +++ 3. Cc bi tp dnh cho bn c t gii:( ) ( )2 2 221 2 322 2 2 21 1 1d d 5; ; d23 1 5 2 3 2 3 1x x xH H H xxx x x x x x+ + + + + IX. Dng 9:( )( )2 2mx + n dxI =ax + b cx + d1. Phng php: ( ) ( )2 2 2 2xdx dxI m n mG nHax b cx d ax b cx d + ++ + + + 2. Cc bi tp mu minh ha:( )( )( ) [ ]( ) ( )32 224 1 71 5 3 1 2x dxx x + ] + ] 312 224x + 3 dxI =x - 2x - 4 3x - 6x + 5 ( )( ) ( ) ( )2 2 22 2 2 2 2 21 1 1du udu du4u 74 7 4J 7Lu 5 3u 2 u 5 3u 2 u 5 3u 2+ + + + + Xt ( )22 215 3 2uduJu u +. t 22 223 23 3t tdtt u u udu + ( )( )142 14 14222 21 5 55udu tdt dt 1 t 17J ln2 17 t 17 t 17t 17 tu 5 3u 2 + + ( ) ( )( ) ( )17 14 17 51 17 14 17 5 1ln ln ln2 17 17 14 17 5 2 1717 14 17 5 +| ` + ++ . ,Xt ( )22 215 3 2duLu u +. t 2 2 2 2 2223 2 3 23ut u ut u ut + + ( )( )( )( )222 22222tdt t 3 2tdt du udu dtuduu ut t 32t t 33u 2t 3 +. Khi :10( )( )14 2 14 2 222 221 2 22du dt dtL217 5tu 5 3u 25 t 3t 3 | ` + . ,

14 221 1 17 t 5ln5 2 17 17 t 5+ ( ) ( )( ) ( )1 70 2 17 2 5 17ln2 8570 2 17 2 5 17+ + ( )( )( ) ( )( ) ( )( ) ( )117 14 17 54 7 70 2 17 2 5 17I 4J 7L ln ln2 17 2 8517 14 17 5 70 2 17 2 5 17 ++ + +( )( )( ) [ ]( ) ( )6 12 22 12 1 11 5 2 1 3x dxx x+ ]+ + + ] 6 -122 22 -12x + 1 dxI =x + 2x + 6 2x + 4x - 1 ( )( ) ( ) ( )6 6 62 2 2 2 2 22 2 22u 1 du udu du2 2J Lu 5 2u 3 u 5 2u 3 u 5 2u 3 + + + Xt ( )62 225 2 3uduJu u+ . t 22 232 32 2t tdtt u u udu+ ( )( )6 3 3222 21 1 2udu tdt dt 2 3 1J arctg arctgt 13 13 13 13 t 13 tu 5 2u 3| ` ++ . , + Xt L( )62 225 2 3duu u + . t 2 2 2 2 2232 3 2 32ut u ut u ut ( )223tdtudu2 t ( )( )( )222223tdt 2 t du udu dtu ut 2 t3t 2 t2u 3 . Khi :( )( )3 6 3 6 3 6 622 2222 1 2 1 2 1 22du dt dt 1 dtL133 513 5tu 5 2u 3t5 2 t52 t | ` + + . , 3 61 213 5 t1 1 1 78 3 5 26 5ln ln ln52 13 5 13 5 t 2 65 78 3 5 26 5| ` ++ +. ,( )( )( )( )24 3 1 1 78 3 5 26 5I 2J L arctg arctg l n13 13 13 2 6578 3 5 26 5+ + | ` . ,BI 3. BIN I V I BIN NNG CAO TCH PHN HM PHN THC HU TI. DNG 1: TCH CC MU S CHA CC NHN T NG BC11Cc bi tp mu minh ha:( )( )1dxA =x 2 x + 5( ) ( )( ) ( )1 5 2 1 1 1 1 2ln7 2 5 7 5 5 7 5x x xdx dx cx x x x x+ | ` + + + + . , ( ) ( )( )( ) ( )( ) ( )( )1 x 4 x 5dx9 x 5 x 2 x 4+ + + 2dxA =x 5 x + 2 x + 4( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )1 1 1 1 2 5 1 4 29 5 2 2 4 63 5 2 18 2 41 1 1 1 1 1 1 5 1 4ln ln63 5 2 18 4 2 63 2 18 2x x x xdx dx dxx x x x x x x xx xdx dx cx x x x x x+ + + ] ] + + + + + + ] +| ` | ` + + + + + + + + . , . , II. DNG 2: TCH CC MU S CHA CC NHN T KHNG NG BC1. Cc bi tp mu minh ha:( )( )( )( )2 222 22 222 2dx 1 x x 3 1 xdx dxdx3 3 xx 3x x 3 x x 31 1 d x 3 dx 1 1 1 x 3ln x 3 ln x c ln c3 2 x 3 2 6x 3 x | ` . , ] | ` + + ]. , ] 13dxB =x 3x( )( )( )4 44 33 4 3 4dx 1 x x 10 1 xdx dxdx10 10x 10 xx x 10 x x 10 | ` . , 27 3dxB =x 10x( )( )2 22 3 2 221 1 d x dx 1 1 x 10 1ln c10 2 20x x 10 x 10x 10| ` | ` + + +. , . , 2. Cc bi tp dnh cho bn c t gii:1 2 3 4 53 9 4 11 5 6 7dx dx dx dx dxB ; B ; B ; B ; Bx 5x x 7x x 8x x 9x x 13x + + + 6 7 83 2 3 2 4 3 2dx dx dxB ; B ; Bx 6x 19x 22 x 3x 14x 12 x 4x 6x 7x 4 + + + + + + + + III. DNG 3: K THUT NHY TNG LU KHI MU S L HM A THC BC 4( ) ( )( ) ( )( ) ( )2 22 2 2 2dx 1 x 1 x 1 1 x 1 1dx ln arctgx c2 4 x 1 2x 1 x 1 x 1 x 1+ ++ + + 14dxC =x 1( )( ) ( )( )2 222 2 22 21 d x 1 1 1 1 x 1d x ln c2 4 4x 1 x 1 x 1x 1 x 1 | ` + + + . , + 24xdxC =x 1( ) ( )( ) ( )2 22 22 22 21 x 1 x 1 1 1 1dx dx2 2 x 1 x 1x 1 x 11 dx 1 dx 1 x 1 1ln arctgx c2 2 4 x 1 2 x 1 x 1+ + | ` + + . , + + + ++ + 234x dxC =x 112( )4441 d x 1 1ln x 1 c4 4x 1 + 344x dxC =x 1( )414 4x 1 1 dx 1 x 1 1dx dx x C x ln arctgx c4 x 1 2x 1 x 1 + + + + ++ 454x dxC =x 1( )( )( )22221 d x 1arctg x c2 2x 1 ++ 64xdxC =x + 1( )4441 d x 1 1ln x 1 c4 4x 1+ + ++ 374x dxC =x + 1( )( )( )( )( )2222211 11d x x 21x xxdx ln c112 21x 2xx 2xxx+ + ++ +++ 284x 1C = dxx + 1( )( )( )2222 22111d x1 x 1xxdx arctg c12 x 21xx 2xx+ ++ + 294x + 1C = dxx + 1( ) ( )( )2 2 2 24 4 42 29 821 x 1 x 1 1 x 1 x 1dx dx dx2 2x 1 x 1 x 11 1 1 x 1 1 x x 2 1C C arctg ln c2 22 x 2 2 2 x x 2 1| `+ + + + + . ,| ` + + + +. , 104dxC =x + 1( ) ( )( )2 2 2 24 4 42 29 821 x 1 x 1 1 x 1 x 1dx dx dx2 2x 1 x 1 x 11 1 1 x 1 1 x x 2 1C C arctg ln c2 22 x 2 2 2 x x 2 1| `+ + + + + + + . ,| ` + + + + + +. , 2114x dxC =x + 1( )4 2 24 2x 1 1 1 1 x 1 1 x x 2 1dx x arctg ln c2x 1 2 x 2 2 2 x x 2 1| `+ + + + + +. , 4124x dxC =x + 1( )( )( )( ) ( )( ) ( )222222 2111 dxd xx x1 11 1x 5 x 4x 5 x 6xx x xd u du 1 1 1 1 x 6x 1du ln c7 u 6 u 1 7 u 6 u 1u 5u 6 x x 1| `+ . , + + + + + | ` + + + . , + + 2134 3 2x - 1 dxC =x 5x 4x 5x + 1( ) ( )2 2 2 24 2 4 2 4 21 x 1 x 1 1 x 1 x 1dx dx dx2 2x x 1 x x 1 x x 1 ]+ + ]+ + + + + +] 144 2dxC =x + x + 113( )( )( )( )2 22 22 22 21 11 11 dx 1 dxd x d x1 1x x x x1 1 2 41 1x 1 x 1x 3 x 1x x x x ]| ` | ` ]+ + ] ]. , . , ] ]| ` | ` ]+ + + +] + + ]] . , . , ]] 2 221 1x x 11 1 1 x 1 1 x x 1x xarctg ln c arctg ln c14 4x x 1 2 3 3 2 3 x 3x 1x + + + ++ ++ +IV. DNG 4: K THUT NHY TNG LU KHI MU S L HM A THC BC 3( )( )( )( ) ( ) ( )2 2dx d x 1x 1 x x 1x 1 x 1 3 x 1 3 ] + + + + ] 13dxD =x 1( )( ) ( )( )( )2 222 2dt 1 t 3t 3 t 3t 1 dt t 3 dtdt3 3 tt 3t 3t t 3t 3 t t 3t 3| ` + + + + + + . , + + + + ( )2 21 dt 1 2t 3 dt 3 dt3 t 2 2t 3t 3 t 3t 3| ` + + + + + . , 221 x 2x 1 1 2x 1ln arctg c6x x 1 2 3 3 + + ++ +( ) ( )( )( ) ( ) ( )2 2dx d x 1x 1 x x 1x 1 x 1 3 x 1 3+ ]+ ++ + + + ] 23dxD =x + 1( )( ) ( )( )( )2 222 2dt 1 t 3t 3 t 3t 1 dt t 3 dtdt3 3 tt 3t 3t t 3t 3 t t 3t 3| ` + + . , + + ( )( )22 21 dt 1 d t 3t 3 3 dt3 t 2 2 3 t 3t 33t24| ` + + + + . , 2 22 21 1 t 2t 3 1 x 2x 1 1 2x 1ln 3arctg c ln arctg c3 2 6t 3t 3 x x 1 3 2 3 3| ` + + + + + + + +. ,( ) ( )( ) ( )( ) ( )222 2xdx 1 x x 1 x 1dx3x 1 x x 1 x 1 x x 1+ + + + + + 33xdxD =x 1

21 1 x 1dx3 x 1x x 1 | ` + + . ,( )( )2 221 dx 1 2x 1 dx 3 dx3 x 1 2 2x x 131x2 2 ] + + ]+ +| ` ]+ + ]. , ]

21 1 2x 1ln x 1 ln x x 1 3arctg c3 23+ ] + + + + ] ]( )( )( ) ( )( ) ( )222 2xdx 1 x x 1 x 1dx3x 1 x x 1 x 1 x x 1 + + + + + + 43xdxD =x + 1( )( )2 2 221 1 x 1 1 dx 1 2x 1 dx 3 dxdx3 x 1 3 x 1 2 2x x 1 x x 131x2 2 ] + | ` ]+ + + + . ,| ` ] + ]. , ] 142221 1 2x 1 1 x 2x 1 1 2x 1ln x 1 ln x x 1 3arctg c ln arctg c3 2 6 x x 1 3 3 3 + + ] + + + + ] + ]V. DNG 5: K THUT NHY TNG LU KHI MU L HM A THC BC 6( )( )( )1 23 33 3dx 1 dx dx 1D D2 2x 1 x 1x 1 x 1 ] ] +] + 16dxE =x 1

( ) ( )( ) ( )2 22 22 22 21 1 x 2x 1 1 2x 1 1 x 2x 1 1 2x 1ln arctg ln arctg2 6 6x x 1 x x 1 2 3 3 2 3 31 x 2x 1 x x 1 1 2x 1 2x 1ln arctg arctg c124 3 3 3 x 2x 1 x x 1 ]| ` | ` + + + + +] + + + ]. , . , ] + + + | ` + + + + + + . ,( )( )213 321 d x 1 du 1D2 2 2u 1x 1 26xdxE =x 1

2 4 2 22 4 21 1 u 2u 1 1 2u 1 1 x 2x 1 1 2x 1ln arctg c ln arctg c2 6 12u u 1 x x 1 2 3 3 2 3 3 ] + + + + + + ]+ + + + ]( )3 3 36 3 31 d x 1 1 x 1 1 x 1ln c ln c3 3 2 6x 1 x 1 x 1 + + + + 236x dxE =x 1( )( ) ( )2 26 321 x d x 1 udu 1 udu2 2 2x 1 u 1u 1 u u 1 + + 346x dxE =x 1 ( )24 2 22 4 21 u 1 1 2u 1 1 x 2x 1 1 2x 1ln arctg c ln arctg c12 12u u 1 x x 1 2 3 3 2 3 3 + + + + + + ++ + + +( ) ( )( ) ( )4 2 22 4 2 62 4 2x x 1 x 1 2 dx dx dxdx 2x 1 x x 1 x 1x 1 x x 1+ + + + + + 456x dxE =x 1 ( ) ( )( ) ( )2 2 22 21 x 2x 1 x x 1 1 2x 1 2x 1 x 1ln arctg arctg arctg c122 3 3 3 x 3 x 2x 1 x x 1| ` + + + + + + + + + +. ,( )6661 d x 1ln x 1 c6 6x 1 + 566x dxE =x 1( )616 6x 1 1 dxdx dx x Ex 1 x 1 + + + 676x dxE =x 1

( ) ( )( ) ( )2 22 21 x 2x 1 x x 1 1 2x 1 2x 1x ln arctg arctg c124 3 3 3 x 2x 1 x x 1 + + + | ` + + + + + + + . ,( ) ( )( ) ( )( )2 2 224 22 4 22211 dxx 1 x 1 dx x 1 dxx1x x 1x 1 x x 1x 1x| ` + . , | ` ++ ++ . , 486x 1E = dxx + 115( )( )( )22 2211d xx 31 1 x x 3 1xxln c ln c12 3 2 3 x x 3 11 x 3x 3xx++ + + ++ ++ ++ ( )( )( )4 2 2 22 62 4 2x x 1 x dx x dxdxx 1 x 1x 1 x x 1 + + ++ ++ + 496x + 1E = dxx + 1( )( )332 6dx 1 d x 1arctgx arctg x c3 3x 1 x 1 + + ++ + ( ) ( )( )( )4 49 862321 x 1 x 1 1dx E E2 2 x 11 1 1 x x 3 1arctgx arctg x ln c2 32 3 x x 3 1+ +| ` + + + + +. , 106dxE =x + 1( ) ( ) ( )3 2 326 6 61 d x 1 d x 1 d x 1D3 2 3 2x 1 x 1 x 1 + ++ + + 2116x + xE = dxx + 1 (thay x2 vo D2) ( )4 2 234 21 1 1 x 2x 1 1 2x 1arctg x ln arctg c3 2 6x x 1 2 3 3| `+ + + + + +. ,VI. DNG 6: S DNG KHAI TRIN TAYLOR a thc Pn(x) bc n c khai trin Taylor ti im xa l:( ) ( )( )( )( )( )( )( )( )n2 nn n nn nP a P a P aP x P a x a x a x a1! 2! n! + + ++ 1. Cc bi tp mu minh ha:( ) 4 31503x 5x + 7x 8F = dxx + 2. t ( )4 34P x 3x 5x 7x 8 + ( ) ( )( )( )( )( )( )( )( )( )( )( )3 42 3 44 4 4 44 4P 2 P 2 P 2 P 2P x P 2 x 2 x 2 x 2 x 21! 2! 3! 4! + + + + + + + +( ) ( ) ( ) ( ) ( )2 3 44P x 66 149 x 2 48 x 2 29 x 2 3 x 2 + + + + + +( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 3 415050 49 48 47 4649 48 47 46 4566 149 x 2 48 x 2 29 x 2 3 x 2F dxx 266 x 2 149 x 2 48 x 2 29 x 2 3 x 2 dx66 149 48 29 3c49 x 2 48 x 2 47 x 2 46 x 2 45 x 2 + + + + + + + ] + + + + + + + ] + + ++ + + + +VII. DNG 7: KTHUT NHY TNG LU KHI MU L HM A THCBC CAO1. Cc bi tp mu minh ha:16( )( )( )99 99 989999 991 3 5 3 1 35 53 53 5 3 5 ]+ ]+] + + 1100dxG =3x + 5xdx x x dx x dxdxxxx x x x

( )99 999999 991 dx 1 d 3x 5 1 1 1 xln x ln 3x 5 c ln c5 x 99 5 99 4953x 5 3x 5 ]+] + + + ] ] ] + +] ( )( )( )( )( )( )( )( ) ( )( )50 50 492 25050 5050 50 49 49 492 50 25050 5050501 2x 7 2x 1 dx 2x dxdx7 7x 2x 7x 2x 7 2x 71 1 2x 7 2x 2x dx 1 dx 2x dx 1 2x dxdx7 7 49 x 72x 7x 2x 72x 7 2x 71 dx 1 d 2x 749 x 502x 7 ]+ ]+ ] + + ] ] ]+ ] ]+] + ]+ + ] ]+

+ 2250dxG =x 2x + 7( )( )( ) ( )5025050505050 501 d 2x 73502x 71 1 1 1 x 1ln x ln 2x 7 ln c49 49.50 49.502x 7350 2x 7 350 2x 7+]]+ + + + +++ +( )( )( ) ( )( )( )n n nk k 1 kn n n1 ax b ax 1 dx 1 d ax bdxb b nbx ax b x ax b ax b+ + + + + 3kndxG =x ax + b ( )( )( )( )( )n n2 k 2 2 k 1 kn n n1 dx 1 d ax b 1 d ax bnbb nbx ax b ax b ax b + + + + + ( ) ( )( )( ) ( )( )nk k 1 kk 1 nnnk n k 1k 1 nn1 1 1 1 1ln x ln ax b cn b bb ax bb k 1 ax b1 x 1 1 1ln cn nb ax bb ax bb k 1 ax b ] + ++ + + ]+ + ] ] + ++ + ]++ + ]( )( )( )( ) ( )( )( )2000 2000 19992000 20002000 10002000200020001 x 2x dx 2x dxdxxx 1 x 1 xdx 1 d 1 x 1 xln x ln1 x c ln cx 1000 1000 1 x1 x + + ++ + + +++ 2000420001 x dxG =x 1+ x( ) ( )( )( )( )( )( )( ) ( )( )( )10 9 10 10 10102 2 210 10 1010 101010 210101 .10 1 1 3 3310 10 103 3 31 3 3 1 33 ln 310 10 310 33+ ++ + + ]+ + + + + ]++ ]+ ] 195210x dxG = =3+ xx x dx x d x xd xx x xd x d xx cxxx( ) ( )( )( )( )( )( )( )( ) ( ) ( )( )( ) ( )50 49 50507 750 5050 506 7 5 650 50 50 5050 506 650 50x .x dx 1 2x 3 3d 2x 32002x 3 2x 31 d 2x 3 d 2x 3 1 1 13 c200 2002x 3 2x 3 5 2x 3 2 2x 31 2 2x 3 5 1 4xc c20010 2x 3 2000 2x 3 + ] ] + + + ] ] ] ]] + + + 996750x dxG =2x 317( ) ( )n n 1knx x dxax b+ 2n-17knx dxG =ax + b( )( )( )nn2 kn1 ax b bd ax bnaax b+ ++

( )( )( )( ) ( ) ( ) ( ) ( )n n2 k 1 k 2 k 2 k 1n n n n1 d ax b d ax b 1 1 bb cna naax b ax b k 2 ax b k 1 ax b ]+ + ] + + ] ] ]+ + + + ]] ( ) ( ) ( )( )( )( ) ( ) ( ) ( )n n2 k 1 k 1n 2 n1 b k 2 k 1 ax b kax bc cnak 1 k 2 ax b na k 1 k 2 ax b + + + + +2. Cc bi tp dnh cho bn c t gii:5****1 2 3 4 58 8 8 8 8xdx x x dx xdx dxG ; G dx ; G ; G ; Gx 1 x 1 x 1 x 1 x 1 + + + VIII. DNG 8: KTHUT CHNG NH THC( )( ) ( )1023x 5 dxx 2x 2 | ` + . ,+ 101123x 5H = dxx + 2 10 111 3x 5 3x 5 1 3x 5d c11 x 2 x 2 121 x 2 | ` | ` | ` + + + + . , . , . ,( )( ) ( )99 9927x 1 dx 1 7x 1 7x 1d2x 1 9 2x 1 2x 12x 1 | ` | ` | ` + + + . , . , . ,+ 9921017x 1H = dx2x + 1 100 1001 1 7x 1 1 7x 1c c9 100 2x 1 900 2x 1 | ` | ` + + + + . , . ,( ) ( )( )( )( )( ) ( )5 5 6 28dx 1 1 dxx 3 x 3 x 5 x 5x 5x 5 x 5 + + + +++ + 35 3dxH =x + 3 x + 5 ( )( ) ( )( )( )667 5 7 51 1 x 3 x 5 1 1x 3d u 1 dux 5x 52 2 ux 3x 5 ] + ++ ]++]++ 6 5 4 3 27 57 2 3 4 527 2 3 41 u 6u 15u 20u 15u 6u 1du2 u115 20 15 6 1u 6 duu2 u u u u1 u20 15 2 16u 15ln u cu22 2u u 4u + + +| ` + + + . ,| ` + + + + . , ( ) ( )( ) ( ) ( ) ( )272 3 471 1 x 3x 3 x 36 15lnx 5 x 52 x 5 21x 5 15 x 5 x 5 x 5 120 2 cx 3 2 x 3 x 3 4 x 32 ]++ + + + ]+ ++ ] ]+ + + ++ + + ]+ + + + ]18Cc bi tp dnh cho bn c t gii:( ) ( )17 3dxH =3x 2 3x + 4; ( ) ( )1 23 4dxH =2x 3x - 1; ( ) ( )35 4dxH =3x + 2 4x - 1BI 4. TCH PHN C BN CA CC HM S LNG GICA. CNG THC S DNG1. KHAI TRIN NH THC NEWTON( )0 1 1 1 1nn n k n k k n n n nn n n n na b Ca Ca b ... Ca b ... C ab Cb + + + + + + +trong ( )!! !knnCk n k v ( )1 2 1 m! . .... m m vi qui c 0!12. CC CNG THC NGUYN HM LNG GIC ( ) ( ) ( ) ( )( )( )( )( )2 21 11 1cos ax bdx sinax b c sinax bdx cos ax b ca adx dxtg ax b c cotg ax b ca acos ax b sin ax b+ + + + + + + + + ++ + B. CC DNG TCH PHNI. Dng 1: ( ) ( ) n n1.1 1.2A = sinx dx ; A cosx dx1. Cng thc h bc2 2 3 31 2 1 2 3 3 3 32 2 4 4cos x cos x sin x sin x cos x cos xsin x ; cos x ; sin x ; cos x + + + 2. Phng php2.1.Nu n chn th s dng cng thc h bc2.2.Nu n3 th s dng cng thc h bc hoc bin i theo 2.3.2.3.Nu 3 n l (n2p +1) th thc hin bin i:( ) ( ) ( ) ( ) ( )221ppsin x sin xdx cos x d cos x n 2p+11.1A = sinx dx = sinx dx( ) ( ) ( ) ( ) ( )( )( )( )( )k pk p 0 1 2 k 2 p 2p p p pk p2k 1 2p 1 0 1 3 k pp p p pC C cos x ... 1 C cos x ... 1 C cos x d cos x1 1 1C cos x C cos x ... C cos x ... C cos x c3 2k 1 2p 1+ + ] + + + + ] ] ] + + + + +]+ +] ]( ) ( ) ( ) ( ) ( )221ppcos x cos xdx sin x d sin x n 2p+11.2A = cosx dx = cosx dx19( ) ( ) ( ) ( ) ( )( )( )( )( )k pk p0 1 2 k 2 p 2p p p pk p2k 1 2p 10 1 3 k pp p p pC C sin x ... 1 C sin x ... 1 C sin x d sinx1 1 1C sinx C sin x ... C sinx ... C sinx c3 2k 1 2p 1+ + ] ++ ++ ] ] ++ ++ +]+ + ]( )3321 22cos xcos x dx dx+| ` . , 61A = cos xdx = ( ) ( )( )3 2 31 11 cos 2x dx 1 3cos 2x 3cos 2x cos 2x dx4 41 3 1 2cos 4x cos 3x 3cos x1 3cos 2x dx4 2 41 17x 6sin 2x 3sin 4x sin 3x 3sin x c16 3 + + + +| ` + + + + + . ,| ` + + + + + . , ( ) ( ) ( ) ( ) ( )48215 5 1 5 55sin x sin x dx cos x d cos x 92A = sin5x dx[ ] ( )2 4 6 83 5 7 911 4cos 5x 6cos 5x 4cos 5x cos 5x d cos 5x51 4 6 4 1cos 5x cos 5x cos 5x cos 5x cos 5x c5 3 5 7 9 + +| ` + + + . ,II. Dng 2:m nB = sin x cos x dx (m, nN)1. Phng php:1.1. Trng hp 1: m, n l cc s nguyna. Nu m chn, n chn th s dng cng thc h bc, bin i tch thnh tng.b. Nu m chn, n l (n 2p +1) th bin i:( ) ( ) ( ) ( ) ( ) ( ) ( )221pm p msin x cos x cos xdx sin x sin x d sin x m 2p+1B = sinx cosx dx( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )( )( )( )k pm k p 0 1 2 k 2 p 2p p p pm 1 m 3 2k 1 m 2p 1 mk p 0 1 k pp p p psin x C C sin x ... 1 C sin x ... 1 C sin x d sin xsin x sin x sin x sin xC C ... 1 C ... 1 C cm 1 m 3 2k 1 m 2p 1 m+ + + + + + ] + + + + ] ] ] + + + + +]+ + + + + +] ]c. Nu m chn, n l (n 2p +1) th bin i:( ) ( ) ( ) ( ) ( ) ( ) ( )221pn p ncos x sin x sin xdx cos x cos x d cos x 2p+1 nB = sinx cosx dx( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )( )( )( )k pn k p 0 1 2 k 2 p 2p p p pn 1 n 3 2k 1 n 2p 1 nk p 0 1 k pp p p pcos x C C cos x ... 1 C cos x ... 1 C cos x d cos xcos x cos x cos x cos xC C ... 1 C ... 1 C cn 1 n 3 2k 1 n 2p 1 n+ + + + + + ] + + + + ] ] ] + + + + +]+ + + + + +] ]d. Nu m l, n l th s dng bin i 1.2. hoc 1.3. cho s m l b hn.201.2. Nu m, n l cc s hu t th bin i v t usinx ta c: ( ) ( ) ( )1 12 22 21n mmm n mB sin x cos xdx sin x cos x cos xdx u u du (*) Tch phn (*) tnh c 1 trong 3 s 1 12 2 2m n m k; ;+ + l s nguyn 2. Cc bi tp mu minh ha( ) ( ) ( ) ( )2 2 124sin x cos x dx 2 41B = sinx cosx dx( ) ( ) ( )( )( )1 11 cos 4x 1 cos 2x dx 1 cos 2x cos 4x cos 2x cos 4x dx16 161 11 cos 2x cos 4x cos 6x cos 2x dx16 21 1 sin 2x sin 4x sin 6x2 cos 2x 2cos 4x cos 6x dx 2x c32 32 2 2 6 + + ] + + ] ]| ` + + + . , ( ) ( ) ( ) ( )111 85 5 5 cos x sin x sin x dx 9 1112B = sin5x cos5x dx( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )411121112 4 6 8112 114 116 118 1201cos 5x 1 cos 5x d cos 5x51cos 5x 1 4cos 5x 6cos 5x 4cos 5x cos 5x d cos 5x51 cos 5x 4 cos 5x 6 cos 5x 4 cos 5x cos 5xc5 112 114 116 118 120 + + ] + + + ] ]( )( ) ( ) ( ) ( ) ( )4 43625 51cos 3 sin 3 sin 3 cos 3 1 cos 3 cos 33x x xdx x x d x 735 4sin3xB = dxcos 3x ( ) ( ) ( )( ) ( ) ( ) ( )42 4 6531 1 11 215 5 5 51cos 3x 1 3cos 3x 3cos 3x cos 3x d cos 3x31 15 15 55 cos 3x cos 3x cos 3x cos 3x c3 11 21 31 + ] + + ] ]( ) ( )( )33 3 2 281 1 dx dxtg x cos x cos xsin xcos xcos x| ` . , 43 5dxB =sinx cosx

( )( )( ) ( )322 4 63 31 tg x1 3tg x 3tg x tg xd tg x d tg xtg x tg x++ + +

( ) ( )3 3 2 423 1 3 1tg x 3 tg x tg xd tg x 3ln tg x tg x tg x ctg x 2 42 tg x ]+ + + + + + + ] ]21( )( )( )( )( )4 44 24 2 4 2cos sin 1 sin sinsinsin cossin 1 sin sin 1 sinxdx d x x xd xx xx x x x + 54dxB =sin xcosx( )( )( )24 2 31 sin x d sinx 1 1 1 1 sinxd sinx ln csinx 2 1 sinx3 sinx sin x 1 sin x+ + + + + ( ) ( ) ( ) ( )5 1 5 43 3 3 3sin cos sin cos cos d x x dx x x x x 63 5dxB =sin xcosx( ) ( ) ( )( )25 223 5 42 33 33 321 usin x cos x d sin x u 1 u du u duu | ` . , t 2321 uvu 3 22u du 3v dv ; ( )1 3 1 32 2232 21 u cos xv tg xu sin x| ` | ` . , . , ( )223 233621 u 3 3 3B u du dv v c tg x c2 2 2u| ` + + . , Cch 2: ( )( ) ( ) ( )5 23 372531 dx 3B tg x d tg x tg x c2cos xsin xcos x + III. Dng 3: ( ) ( ) n n3 . 1 3 . 2C = tg x dx ; C = cotg x dx (nN)1. Cng thc s dng( ) ( )221dxtg x dx d tg x tg x ccos x+ + ( ) ( )221dxcotg x dx d cotg x cotg x csin x+ + ( )sin x d cos xtg xdx dx ln cos x ccos x cos x + ( )cos x d sin xcotg xdx dx ln sin x csin x sin x + 2. Cc bi tp mu minh ha ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 4 2 62 2 2tg 1 tg tg 1 tg tg 1 tgk k kx x x x x x + + + + 2k1C = tgx dx( ) ( )( )( ) ( )( )k 1 k 2k 8 02 2tg x 1 tg x ... 1 tg x 1 tg x 1 dx ] + + + + + ]( ) ( ) ( )( )( ) ( )( )( ) ( ) ( )( ) ( )k 1 k 2k 2 2k 4 2k 6 02k 1 2k 3 2k 5k 1 ktg x tg x tg x ... 1 tg x d tg x 1 dxtg x tg x tg x tg x1 1 x c2k 1 2k 3 2k 5 1 ] + + + ] + + + + ( ) ( ) ( ) ( ) ( )2 1 2 32 2tg 1 tg tg 1 tgk kx x x x + + + 2k+12C = tgx dx( ) ( )( )( ) ( )( )k 1 k 2k 52 2tg x 1 tg x ... 1 tg x 1 tg x 1 tg x dx ]+ + + + + ]22( ) ( ) ( )( )( ) ( )( )( ) ( ) ( )( )( )( )k 1 k 2k 1 2k 3 2k 52k 2k 2 2k 4 2k 1 ktg x tg x tg x ... 1 tg x d tg x 1 tg xdxtg x tg x tg x tg x1 1 ln cos x c2k 2k 2 2k 4 2 ] + + + ] + + + ( ) ( ) ( ) ( ) ( )2 2 2 42 2cotg 1 tg cotg 1 tgk kx co x x co x + + + 2k3C = cotgx dx( ) ( )( )( ) ( )( )k 1 k 2k 6 02 2cotg x 1 co tg x ... 1 cotg x 1 co tg x 1 dx ]+ + + + + ]( ) ( )( )( ) ( )( )( ) ( ) ( )( ) ( )k 1 k 2k 2 2k 4 02k 1 2k 3 2k 5k 1 kcotg x cotg x ... 1 cotg x d cotg x 1 dxcotg x cotg x cotg xcotg x1 1 x c2k 1 2k 3 2k 5 1 ] + + + ] ] ] + + + + ] ( ) ( ) ( ) ( ) ( )2 1 2 32 2cotg 1 tg cotg 1 tgk kx co x x co x + + + 2k+14C = cotgx dx( ) ( )( )( ) ( )( )k 1 k 2k 5 12 2cotg x 1 co tg x ... 1 cotg x 1 co tg x 1 cotg x dx ]+ + + + + ]( ) ( )( )( ) ( )( )( ) ( )( )( )( )k 1 k 2k 1 2k 32k 2k 2 2k 1 kcotg x cotg x ... 1 cotg x d cotg x 1 cotg x dxcotg x cotg x cotg x1 1 ln sin x c2k 2k 2 2 ] + + + ] ] ] + + + +] ( ) ( ) ( ) ( ) ( )5 4 3 2tg 5 tg cotg 10 tg cotg x x x x x

+ + +

55C = tgx + cotgx dx( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )2 3 4 55 5 3 35 3 5 310 tg x cotg x 5tg x cotg x cotg x dxtg x cotg x 5 tg x 5 cotg x 10 tg x 10cotg x dxtg x 5 tg x 10 tg x dx cotg x 5 cotg x 10cotg x dx]+ + +] ] + + + + + ] ]] + + + + + ]] ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )32 232 23 34 42 2tg x 1 tg x 4 tg x 1 tg x 6 tg x dxcotg x 1 cotg x 4cotg x 1 cotg x 6cotg x dxtg x 4 tg x d tg x 6 tg x dx cotg x 4cotg x d cotg x 6 cotg x dxtg x cotg x2 tg x 6ln cos x 2cotg x 6ln sin x c4 4 ] + + + + ] ]+ + + + + ] ]] + + + + ]] + + + IV. Dng 4:( )( )( )( ) m m 4 . 1 4 . 2n ntg x cotg xD = dx ; D = dxcos x sin x1. Phng php:Xt i din ( )( )4 1m.ntg xD dxcos x1.1. Nu n chn (n2k) th bin i:23( )( )( ) ( ) ( ) ( )1122 211kkm m dxtg x tg x tg x d tg xcos x cos x| ` + . , m4.12ktgxD = dxcosx( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )1 p k 1m0 1 2 p 2 k 1 2k 1 k 1 k 1 k 1m 1 m 3 m 2p 1 m 2k 10 1 p k 1k 1 k 1 k 1 k 1tg x C C tg x ... C tg x ... C tg x d tg xtg x tg x tg x tg xC C ... C ... C cm 1 m 3 m 2p 1 m 2k 1 + + + + + ] + + + + + ] + + + + + ++ + + + + 1.2. Nu m l, n l (m2k +1, n2h+1) th bin i:( )( )( ) ( )2 22221 1h hkk tg x sin xtg x dx tg x dxcos x cosx cos xcos x| ` | ` . , . , 2k+14 .12h+1tgxD = dxcosx( )k 2hk2 2h21 1 11 d u 1 u ducos x cos xcos x| ` | ` | ` . , . ,. , ( y 1ucos x )( ) ( ) ( ) ( ) ( )k k 1 k pp k2h 0 2 1 2 p 2 kk k k ku C u C u ... 1 C u ... 1 C du ] + + + + ]( ) ( )2k 2h 1 2k 2h 1 2k 2h 2p 1 2h 1p k0 1 p kk k k ku u u uC C ... 1 C ... 1 C c2k 2h 1 2k 2h 1 2k 2h 2p 1 2h 1+ + + + + + + + + + ++ + + + + +1.3. Nu m chn, n l (m2k, n2h+ 1) th s dng bin i:( )( )( )( )( )( )( )( ) ( )( )( )( ) ( ) ( )2k2k 2k4.12h 1 k h 1 2 k h 122k 2k 2 2 2k 2 2k 24.1k h 1 k h 1 k h 1 k h2 2 2 2tg xsin x cos x sin xD dx dx d sin x ; u s inxcos x cos x1 sin xu du u 1 1 u u du u duD du1 u 1 u 1 u 1 u+ + + + + + + + + + + + ] ] H thc trn l h thc truy hi, kt hp vi bi tch phn hm phn thc hu t ta c th tnhc D4.1.2. Cc bi tp mu minh ha:( )( )( )( ) ( )( ) ( ) ( )227 722 21 1tg 3 tg 3 1 tg 3 tg 33cos 3 cos 3dxx x x d xx x ] + ] ] 716tg3xD = dxcos3x( ) ( ) ( ) ( )( ) ( ) ( )8 10 127 2 4tg 3x tg 3x tg 3x 1 1tg 3x 1 2 tg 3x tg 3x d tg 3x 2 c3 3 8 10 12 ] ] + + + + +] ] ]( )( )( )( ) ( )3102 2155 5dxcotg xsin x sin x ] ] ] 1028cotg5xD = dxsin5x

( ) ( )( ) ( ) ( ) ( )310211 13 15 171cotg5x 1 cotg 5x d cotg5x5cotg5x cotg5x cotg5x cotg5x13 3 c5 11 13 15 17 ] + ] ] ] + + + + ]( )( )( )946 4 144 4tg xtg x dxcos x cos x| ` . , 7395tg4xD = dxcos4x24( )( )( )( ) ( ) ( ) ( )3 94394 22101 99 97 9594 6 4 2101 99 97 951 1 1 1 11 d u u 1 du4 cos 4x cos 4x 4cos 4x1 1 u u u uu u 3u 3u 1 du 3 3 c4 4 101 99 97 951 1 1 3 1c4101 cos 4x 33 cos 4x 97 cos 4x 95 cos 4x ] | ` | ` ]. , . , ] ] + + + ] ] ] + + ] ] ( )( )( )408 3 133 3cotg xcotg x dxsin x sin x| ` . , 9441cotg3xD = dxsin3x( )4 40440 221 1 1 1 11 d u u 1 du3 sin3x sin3x 3sin x| ` | ` | ` . , . ,. , ( )49 47 45 43 41440 8 6 4 21 1 u u u u uu u 4u 6u 4u 1 du 4 6 4 c3 3 49 47 45 43 41 ] + + + + + ] ]( ) ( ) ( ) ( ) ( )49 47 45 43 411 1 4 2 4 1c349 sin 3x 47 sin 3x 15 sin 3x 43 sin 3x 41 sin 3x ] + + + ] ]( ) ( )( ) ( )( )2 22 2 21sinx cosxdx sinxdsinxsinxcosx cosx| ` . , 25tgx dxD =cosx( ) ( )( )( )( ) ( )221 sin x 1 sin x 1 1d sin x d sin x1 sin x 1 sin x 1 sin x 1 sin x ] + | ` ]+ + . , ] ( ) ( )( )2 2 21 1 2 1 1 1 sin xd sin x ln c1 sin x 1 sin x 1 sin x1 sin x1 sin x 1 sin x+] + + ] + + ]( )( )( ) ( )( )( )( )4 44 2 321sin x cos xdx sin xd sin xcos x cos xsin x 46tgxD = dxcosx( )( )( ) ( ) ( )4 4 22 13 3 3 22 2 2 2u du 1 1 u du 1 udu du I I1 u 1 u 1 u 1 u + ( )( )212211u duIu+( )( )22 2 2111 dud u1 uu uc c11 u1 1uuuuuu| `+. , + +| ` . , ( )2321duIu( ) ( )( ) ( )331 1 u 1 u 1 1 1du du8 1 u 1 u 8 1 u 1 u ] + + ] + ] ]+ +] ] ( ) ( ) ( )3 321 1 1 3 1 1du8 1 u 1 u1 u 1 u 1 u ]| ` + + + ] + . , + ]25( ) ( )( )( ) ( )( )( ) ( )( )( )( )( ) ( )2 2 2 22 2 2 2 22 2 221 2 2 2 22 2 21 1 1 du 1 1 u 1 u 1 u 1 u6 3 du8 82 1 u 2 1 u1 2 1 u 1 uu 3 1 u du 3 du u 3 3 1 uI ln c8 8 8 16 1 u 1 u4 1 u 1 u 4 1 u ]+ + + ] + +] ] +] ]]+ + + + + + + u( )( )( )( )( )( )( )6 2 1 1 12222 2 22 2332 42u 3 3 1 uD I I I ln I8 16 1 u4 1 uu 5 u 3 1 u 2u 5u 1 u 3 1 uln c ln c8 16 1 u 16 1 u1 u4 1 u 8 1 u5u 3u 3 1 u 5 sin x 3sin x 3 1 sin xln c ln c16 1 u 16 1 sin x8 cos x8 1 u+ + + + + + + + + + + + + + + 3. Cc bi tp dnh cho bn c t gii:( )( )( )( )( )( )( )( )20 11 4 61 2 3 48 21 3 5tg 6x cotg 3x tg x cotg 2xD dx ; D dx; D dx ; D dxcos 6x sin 3x cos x cos 2x V. Dng 5: S dng cng thc bin i tch thnh tng1. Phng php:

( ) ( ) ( ) ( ) [ ]( ) ( ) ( ) ( ) [ ]( )( ) ( ) ( ) [ ]( )( ) ( ) ( ) [ ]5 15 25 35 412121212....E cos mx cos nx dx cos m n x cos m n xdxE sin mx sin nx dx cos m n x cos m n xdxE sin mx cos nx dx sinm n x sinm n xdxE cos mx sin nx dx sinm n x sinm n xdx + + + + + + 2. Cc bi tp mu minh ha:( )12 14 42cos x cos x cos x + 1E = cos2x .cos5x .cos9x dx( ) ( ) [ ]1 1 sin16x sin12x sin 6x sin 2xcos16x cos12x cos 6x cos 2x dx c4 4 16 12 6 2| ` + + + + + + + . ,( )( )3 384cos x cos xsin x dx+ 32E = cosx sin8x dx( ) ( ) ( )1 1 3 13 cos x sin 8x cos 3x sin 8x dx sin 9x sin 7x sin 11x si n 5x dx4 4 2 21 3 3 1 1cos 9x cos 7x cos11x cos 5x c8 9 7 11 5 ] + + + + ] ]| ` + + + + . , ( ) ( )( ) ( ) ( )2 11 2 13 78cos x sin x sin x dx + 43E = sinx sin3x cos10x dx26

( ) ( )( )( ) ( )211 2cos 2x cos 2x sin13x sin 7x dx81 1 cos 4x1 2cos 2x sin13x sin 7x dx8 213 4cos 2x cos 4x sin13x sin 7x dx16 + ++ | ` + + . , + +

( ) ( ) ( ) [ ]( ) ( )( )13 sin13x sin 7x 4 cos 2x sin13x sin 7x cos 4x sin13x sin 7x dx1613 sin13x sin 7x 2 sin15x sin11x sin 9x sin 5x161sin17x sin 9x sin11x sin 3x dx2 + + + +

+ + + + +

]+ + + +]]

( )1sin17x 4sin15x 6sin13x 3sin11x 3sin 9x 6sin 7x 4 sin 5x sin 3x dx321 cos17x 4 cos15x 6 cos13x 3cos11x cos 9x 6 cos 7x 4 cos 5x cos 3xc32 17 15 13 11 3 7 5 3 + + + | ` + + + + . ,( ) ( ) ( ) ( ) ( )3 2cosx cosx sin5 x dx 54E = cosx sin5x dxcos3x 3cos x 1 cos 2xsin5xdx4 2+ + ( ) ( ) [ ]( ) ( )( ) ( ) ( ) [ ]1cos 3x 3cos x sin 5x cos 3x 3cos x cos 2x sin 5x dx81 sin 7x sin 3xcos 3x 3cos x sin 5x cos 3x 3cos x dx8 212sin 5x cos 3x 3cos x cos 3x 3cos x sin 7x sin 3x dx16 + + ++ ] + + + ] ] + + + +( ) ( ) ( )( ) ( )12 sin8x sin 2x 6 sin 6x sin 4x sin10x sin 4x323 sin8x sin 6x sin 6x 3 sin 4x sin 2x dx

+ + + + + +

]+ + + + +]]( )1sin10x 5sin8x 10sin 6x 10sin 4x 5sin 2x dx321 cos10x 5cos8x 5cos 6x 5cos 4x 5cos 2xc32 10 8 3 2 2 + + + + | ` + + + + + . ,( ) ( ) ( ) ( ) ( ) ( )( )3 4 3 4222sin x sin x sin x sin xdx dxsin x cos xcos x xcos x sin xcosx . sin 2 x + 5sin3x sin4xE = dxtgx + cotg2x27( ) ( ) ( ) ( )1sin2x sin3x sin4x dx cos 2x cos6x sin3x dx2 ( ) ( ) [ ]1 1 cos 5x cos x cos 9x cos3xsin 5x sin x sin 9x sin 3x dx c4 4 5 1 9 3 | ` + + + + . ,3. Cc bi tp dnh cho bn c t gii:( ) ( ) ( ) ( )( )( )54 3 5 21 2 32sin8x dxE sin 3x cos 2x dx ; E sin x cos 5x dx ; Etg3x tg5x + 28