LI NI U Ngy nay php tnh vi tch phn chim mt v tr ht sc quan trng trong Ton hc, tch phn c ng dng rng ri nh tnh din tch hnh phng, th tch khi trn xoay, ncnlitngnghincucagiitch,lnntngcholthuythm,lthuyt phng trnh vi phn, phng trnh o hm ring...Ngoi ra php tnh tch phn cn c ng dng rng ri trong Xc sut, Thng k, Vt l, C hc, Thin vn hc, y hc... Php tnh tch phn c bt u gii thiu cho cc em hc sinh lp 12, tip theo c ph bin trong tt c cc trng i hc cho khi sinh vin nm th nht v nm th hai trong chng trnh hc i cng. Hn na trong cc k thi Tt nghip THPT v k thi Tuyn sinh i hc php tnh tch phn hu nh lun c trong cc thi mn Ton ca khi A, khiB vckhiD.Bncnh , phptnhtch phncng lmt trong nhng ni dung thi tuyn sinh u vo h Thc s v nghin cu sinh. Vitmquantrngcaphptnhtchphn,chnhvthmtivitmtskinh nghimgingdytnhtchphncakhi12vichuynTNHTCHPHN BNGPHNG PHP PHNTCH-I BIN SVTNGPHN phn no cng c, nng cao cho cc em hc sinh khi 12 cc em t kt qu cao trong kthiTtnghipTHPTvkthiTuynsinhihcvgipchoccemcnntng trong nhng nm hc i cng ca i hc. Trongphnnidung chuyndiy, ti xin cnura mtsbi tpminh ha c bn tnh tch phn ch yu p dng phng php phn tch, phng php i bin s, phng php tch phn tng phn. Cc bi tp ngh l cc thi Tt nghip THPT v thi tuyn sinh i hc Cao ng ca cc nm cc em hc sinh rn luyn k nng tnh tch phn v phn cui ca chuyn l mt s cu hi trc nghim tch phn. CHUYN :CC PHNG PHP TNH TCH PHN MC LC Li ni u1 Mc lc2 I. Nguyn hm: I.1. nh ngha nguyn hm3 I.2. nh l3 I.3. Cc tnh cht ca nguyn hm3 I.4. Bng cng thc nguyn hm v mt s cng thc b sung4 II. Tch phn: II.1. nh ngha tch phn xc nh5 II.2. Cc tnh cht ca tch phn5 II.3 Tnh tch phn bng phng php phn tch5 Bi tp ngh 19 II.4 Tnh tch phn bng phng php i bin s10 II.4.1Phng php i bin s loi 110 nh l v phng php i bin s loi 113 Mt s dng khc dng phng php i bin s loi 114 Bi tp ngh s 214 Bi tp ngh s 315 Bi tp ngh s 4: Cc thi tuyn sinh i hc Cao ng16 II.4.2Phng php i bin s loi 216 Bi tp ngh s 521 Cc thi Tt nghip trung hc ph thng22 Cc thi tuyn sinh i hc Cao ng22 II.5.Phng php tch phn tng phn23 Bi tp ngh s 6: Cc thi tuyn sinh i hc Cao ng28 III.Kim tra kt qu ca mt bi gii tnh tch phn bng my tnh CASIO fx570-MS29 Bi tp ngh s 7: Cc cu hi trc nghim tch phn30 Ph lc36 CHUYN :CC PHNG PHP TNH TCH PHN I. NGUYN HM: I.1. NH NGHA NGUYN HM: HmsF(x)cgilnguynhmcahmsf(x)trn(a;b)nuvimi x(a;b): F(x) = f(x) VD1:a) Hm s F(x) = x3 l nguyn hm ca hm s f(x) = 3x2 trn R b) Hm s F(x) = lnx l nguyn hm ca hm s f(x) = 1x trn (0;+) I.2. NH L: Nu F(x) l mt nguyn hm ca hm s f(x) trn (a;b) th: a) Vi mi hng s C, F(x) + C cng l mt nguyn hm ca f(x) trn khong . b)Ngcli,minguynhmcahmsf(x)trnkhong(a;b)ucthvit di dng F(x) + C vi C l mt hng s. Theo nh l trn, tm tt c cc nguyn hm ca hm s f(x) th ch cn tm mt nguyn hm no ca n ri cng vo n mt hng s C.Tp hp cc nguyn hm ca hm s f(x) gi l h nguyn hm ca hm s f(x) v c k hiu: f(x)dx(hay cn gi l tch phn bt nh) Vy: f(x)dx =F(x)+C VD2:a) 22xdx = x +C b)sinxdx =- cosx +C c) 21dx=tgx +Ccos x I.3. CC TNH CHT CA NGUYN HM: 1) ( )f(x)dx f(x)'=2)( ) =a 0 a.f(x)dx af(x)dx3) ( = f(x)g(x) dx f(x)dx g(x)dx 4)( ) ( ) = f(x)dx =F(x)+C f u(x) u'(x)dx F u(x) +CVD3:a)( )4 2 5 3 2-6x + - 2x +4x 5x 8x dx = x +Cb)( ) 2x 6cosx.sinxdx =-6cosx.d cosx=-3cos+C CHUYN :CC PHNG PHP TNH TCH PHN I.4. BNG CNG THC NGUYN HM: BNG CC NGUYN HM C BN NGUYN HM CC HM S CP THNG GPNGUYN HM CC HM S HP ( )( )( ) + +1x xxx2222dx = x +Cxx dx = + C ( -1)+1dx= ln x +C (x 0)xe dx = e +Caa dx = +C 0 < a 1lnacosx dx = sinx +Csinx dx = -cosx +Cdx= 1+ tg x dx = tgx +C(x k )cos x 2dx= 1+cotg x dxsi1/2/3/4/5/6/7/8/x/n9 = -cotgx +C(x k ) ( )( ) +

+1u uuu222du =u+Cuu du = +C( -1)+1du=ln u +C (u =u(x) 0)ue du = e +Caa du = +C0 < a 1lnacosu du = sinu+Csinu du = - cosu+Cdu= 1+tg u du = tgu+C (u k1/2/3/4/5/6/7/8/9/)cos u 2du= 1+csin u( ) 2otg u du = -cotgu+C(u k ) CC CNG THC B SUNG CNG THC NGUYN HM THNG GP: ( )( )( )( ) ( )( ) +1ax+b ax+bkxkx1dx = 2 x + C (x 0)xax +b 1ax +b dx = + C (a 0)a +11 1dx = ln ax +b + C(a 0)ax +b a1e dx = e + C (a 0)aaa dx = + C0 k R, 0 < a 1k.lna1cos ax +b dx = sin ax +b1/2/3/4/5/6/7+ C (a 0)a1sin ax +b dx = - / cosa( ) +ax +b + C (a 0)tgx dx = - ln cosx + C (x k )2cotgx dx = ln sinx + C ( 9/ x/k8) CC CNG THC LY THA:

m n m+nmm-n -nn n1 nn m m m ma. a= aa 1 = a ;1/2/3/= aa aa = a; a= a CC CNG THC LNG GIC: a. CNG THC H BC: ( ) ( )2 21/ 21 1sin x=1- cos2x cos x =1+cos2x2 2/b.CNGTHCBINITCHTHNHTNG ( ) ( )( ) ( )( ) ( ) ( ( ( 1 cosa.cosb=cos a- b +cos a+b21 sina.sinb=cos a- b - cos a+b21 sina.cosb=sin a- b +sin a+b21/2/3/ CHUYN :CC PHNG PHP TNH TCH PHN II. TCH PHN: II.1. NH NGHA TCH PHN XC NH: Gi s hm s f(x) lin tc trn mt khong K, a v b l hai phn t bt k ca K, F(x) l mt nguyn hm ca hm s f(x) trn K. Hiu F(b) F(a) c gi l tch phn t a n b ca f(x). K hiu: baba= f(x)dx =F(x) F(b)-F(a) II.2. CC TNH CHT CA TCH PHN: =( ) 0 / 1aaf x dx= 2/ ( ) ( )a bb af x dx f x dx= b ba ak f x dx k f x dx k . ( ) . ( ) ( 3/ 0) = [ ( ) ( ) 4 ] / ( ) ( )b b ba a af x g x dx f x dx g x dx= + baf(x) ( ) ) 5/ (c ba cdx f x dx f x dxvi c(a;b) 6/ Nu f x x a b ( ) 0, [ ; ] tha( ) 0bf x dx . 7/Nu f x g x x a b ( ) ( ), [ ; ] th a( ) ( )b baf x dx g x dx . 8/Nu m f x M x a b ( ) , [ ; ] th a( ) ( ) ( )bm b a f x dx M b a . 9/ t bin thin trn [ ; ] a b = ( ) ( )taG t f x dxl mt nguyn hm ca( ) f tv= ( ) 0 G a II.3. TNH TCH PHN BNG PHNG PHP PHN TCH: Ch 1: tnh tch phn=( )baI f x dxta phn tch= + +1 1( ) ( ) ... ( )m mf x k f x k f xTrong : =ik i m 0( 1, 2, 3,..., )cc hm=if x i m ( )( 1, 2, 3,..., ) c trong bng nguyn hm c bn. VD4: Tnh cc tch phn sau: CHUYN :CC PHNG PHP TNH TCH PHN

22 3 2-13 2 3 22-1=(3x - 4x+3)dx =(x - 2x+3x)=(2- 2.2+3.2)-((-1)- 2.(-1) +3.(-1)) =121) I Nhnxt:Cu 1 trnta ch cn pdng tnhcht 4 vs dngcngthc 1/ v 2/ trong bng nguyn hm. 2 I2 4 3 2213x -6x+4x - 2x +4)= dxx Nhn xt: Cu 2 trn ta cha p dng ngay c cc cng thc trong bng nguyn hm, trc ht tch phn s trong du tch phn (ly t chia mu) ri p dng tnh cht 4 v s dng cng thc 1/, 2/, 3/ trong bng nguyn hm. I +== 2 2 4 3 222 21 13 2213x -6x+4x - 2x +4 2 4= dx =(3x -6x +4- )dxx x x4(x -3x+4x - 2ln | x |- ) 4- 2ln2x 3) I2 20x -5x +3= dxx +1 Nhnxt:Cu3trntacngchapdngngayccccngthctrongbng nguyn hm, trc ht phn tch phn s trong du tch phn (ly t chia mu) ri p dng tnh cht 4 v s dng cng thc 1/, 2/trong bng nguyn hm v cng thc 3/ b sung. I 6 x| | + |\ | | |\ 2 2 20 0220x -5x +3 9= dx = dxx +1 x +1x = -6x +9ln | x +1| = 2 -12+9ln3 =9ln3 -102 ( )4) I1x -x x -x -x0=e 2xe +5 e -edx Nhn xt: Cu 4: biu thc trong du tch phn c dng tch ta cng cha p dng ngay c cc cng thc trong bng nguyn hm, trc ht nhn phn phi rt gn ri p dng tnh cht 4 v s dng cng thc 1/, 2/, 5/ trong bng nguyn hm. ( ) ( )10 I| | = |\ 1 1 xx -x x -x -x x 20 05 4=e 2xe +5 e -edx = 2x+5-1dx =x+ - xln5 ln5 5)I=44022=(4cosx+2sinx - )dx (4sinx -2cosx - 2tgx) = 22 - 2 - 2+2=2cos x0Nhn xt: Cu 5 trn ta ch cn p dng tnh cht 4 v s dng cng thc 6/, 7/ v 8/ trong bng nguyn hm. CHUYN :CC PHNG PHP TNH TCH PHN 6) I=8080=(4sin2x - 12cos4x)dx (-2cos2x - 3sin4x) = - 2 -3+2 = -1- 2Nhn xt: Cu 6 trn ta cng ch cn p dng tnh cht 4 v s dng cng thc 6/ , 7/ trong bng nguyn hm phn cc cng thc b sung. 7)I1202= sin (2x - )dx4 Nhn xt: Cu 7 hc sinh c th sai v s dng nhm cng thc 2/ trong bng bng nguyn hm ct bn phi, bi xem 2u=sin (2x - )42 (hi ging o hm hm s hp). Vi cu 7 trc ht phi h bc ri s dng cng thc 6/ trong bng nguyn hm phn cc cng thc b sung. ( )I | | |\ | | | | | | |||\ \ \ 12 12 120 0 012021 1= sin (2x - )dx = 1- cos(4x - ) dx = 1- sin4x dx4 2 2 21 1 1 1 1 1= x + cos4x= + cos- 0 + cos0 =- 2 4 2 12 4 3 2 4 24 161 8/I 160= cos6x.cos2xdx Nhn xt: cu 8: biu thc trong du tch phn c dng tch ta cng cha p dng ngay c cc cng thc trong bng nguyn hm, trc ht phi bin i lng gic bin i tch thnh tng ri p dng tnh cht 4 v s dng cng thc 6/ trong bng nguyn hm phn cc cng thc b sung. ( ) I | |= |\ 16 160 01601 1 1 1= cos6x.cos2xdx = cos8x +cos4x dx sin8x + sin4x2 2 8 4 ( )0 0 | || | | | = = = | || |\ \ \ 1 1 1 1 1 1 1 1 2 1sin + sin sin+ sin + 1+ 22 8 2 4 4 2 8 4 2 8 8 16 9)I22-2= x -1dx Nhnxt:Cu9biuthctrongdutchphncchagitrtuyti,tahng hc sinh kh du gi tr tuyt i bng cch xt du biu thc x2 1 trn [-2;2] v kt hp vi tnh cht 5/ ca tch phn kh gi tr tuyt i. CHUYN :CC PHNG PHP TNH TCH PHN ( ) ( ) ( )I5 +| | | | | | + = |||\ \ \ 2 -1 1 22 2 2 2-2 -2 -1 13 3 3-1 1 2-2 -1 1= x -1dx = x -1 dx x -1 dx x -1 dxx x x= - x - x - x3 3 3 10)I3223x +9= dxx - 4x -5 Nhn xt:Cu 10 trnta khng thchinphpchiaa thc c nhcu 2v 3, mtkhcbiuthcdimuphntchcthnh (x -5)(x +1)nntatchbiuthc trongdutchphnnhsau: 23x+9 A B 4 1= + = -x- 4x -5 x -5 x+1 x -5 x+1(phngphphs bt nh) ( )I| | |\ = 3 322 2323x +9 4 1= dx = - dx =4ln | x -5 |-ln | x +1|x - 4x -5 x -5 x +144ln2 -ln4- 4ln3+ln3 = 2ln2 -3ln3 =ln27

Ch 2: tnhI 22a'x +b'= dx(b - 4ac 0)ax +bx +c ta lm nh sau: TH1: Nu 2b - 4ac =0 , khi ta lun c s phn tch 2 2bax +bx +c =a(x + )2a

I 2 2b ba' ba'a'(x + )+b' - b' -a' dx dx2a 2a 2a= dx = +b b ba aa(x + ) x + (x + )2a 2a 2a

TH2: Nu2 21 2b - 4ac >0 ax +bx +c =a(x - x )(x - x) . Ta xc nh A,B sao cho 1 2a'x +b' = A(x - x )+B(x - x) , ng nht hai v 1 2A+B=a'Ax +Bx = -b' I 1 21 2 2 11 A(x - x )+B(x - x) 1 A B= dx = ( + )dxa (x - x )(x - x) a x - x x - x. CHUYN :CC PHNG PHP TNH TCH PHN Ch 3: TH1: tnh I1 2 nP(x)= dx(x -a )(x -a )...(x -a ) ta lm nh sau: 1 2 n1 2 n 1 2 nA A A P(x)= + +...+(x -a )(x -a )...(x -a ) (x -a ) (x -a ) (x -a ) TH2: tnhI = m k r1 2 nP(x)dx(x -a ) (x -a ) ...(x -a ) ta lm nh sau: m k r1 2 nP(x)(x -a ) (x -a ) ...(x -a )=1 2 mm m-11 2 mA A A+ +...+ +...(x - a ) (x - a) (x - a ) TH3: tnh IP(x)= dxQ(x) vi P(x) v Q(x) l hai a thc: * Nu bc ca P(x) ln hn hoc bng bc ca Q(x) th ly P(x) chia cho Q(x). * Nu bc ca P(x) nh hn bc ca Q(x) th tm cch a v cc dng trn. Nhn xt:Vd 4 trn gmnhngbi tptnh tch phn n ginmhc sinh c th p dng ngay bng cng thc nguyn hm gii c bi ton hoc vi nhng php bin i n gin nh nhn phn phi, chia a thc, ng nht hai a thc, bin i tch thnh tng...Quavd4nynhmgipccemthuccng thcvnmvngphptnh tch phn c bn.BI TP NGH 1: Tnh cc tch phn sau: 1)I130=(x x +2x+1)dx2) = 2 2 3212x x +x x - 3x +1dxx 3)I0 3 2-1x -3x -5x +3= dxx - 2 ( )4)I222-2= x+x -3 dx ( )5)I60= sinx +cos2x - sin3xdx 6)I120= 4sinx.sin2x.sin3xdx 7)I0164= cos 2xdx 8)I22-2= x+2x -3dx 9)I421dx=x -5x +6 10)I10dx=x +1+ x

11)I2x+2x +6= dx(x -1)(x - 2)(x - 4)

12) I23x+1= dx(x -1) (x +3) 13)I 4 2xdx=x -6x+5 14)I74 2x dx=(1+x) CHUYN :CC PHNG PHP TNH TCH PHN II.4. TCH PHN BNG PHNG PHP I BIN S: II.4.1. Phng php i bin s loi 1: Tacch(SGKtrang123):Tchphn baf(x)dxchphthucvohmsf(x), cn a v b m khng ph thuc vo cch k hiu bin s tch phn. Tc l: ... = = = b b ba a af(x) f(t) f(u) dx dt du Trong mt s trng hp tnh tch phn m khng tnh trc tip bng cng thc hay qua cc bc phn tch ta vn khng gii c. Ta xt cc trng hp c bn sau: VD5: Tnh cc tch phn sau: 1) I =2220dx2 - x Phntch:Biuthctrongdutchphncchacnbchai,takhngkhcn bng php bin i bnh phng hai v c, ta th tm cch bin i a cn bc hai v dng 2A , khi ta s lin tng ngay n cng thc: 2 2x = x = x 1-sin cos cos, do : t x = 2sint dx = 2costdt ,; ( ( -2 2ti cn: 2 2x = 2sint = t =2 2 6 x =0 2sint =0 t =0I 6 6 662 20 0 00= =2cost.dt 2cost.dt= dt =t =62-2sin t 2(1-sin t) ( v0;( ( cost >06t ) TrongVDtrnkhitathayinhsau:I =220dx2- x.Hcsinhlmtngtv c kt quI2= . Kt qu trn b sai v hm s( ) f x =212-x khng xc nh khi2 x= . Do khi ra dng trn Gio vin cn ch : hm s( ) f xxc nh trn [a;b] CHUYN :CC PHNG PHP TNH TCH PHN 2)I6220= 3 - x dx t x = sint dx = costdt 3 3 ,; ( ( -2 2ti cn: 6 6x = 3sint = t =2 2 4 x =0 2sint =0 t =0( ) | || | ||\ \ 4 4 442 20 0 00. = = 3 3 1 3 1I = 3 -3sin t 3cost.dt 3cos t.dt 1+cos2t .dt = t+ sin2t = +2 2 2 2 4 2a) Khi gp dng 2 22 2dxa - x dxhaya -x(a > 0) tx = sint a. dx =a.cost.dt ,; ( ( -2 2t( bin i a cn bc hai v dng 2A , tc l: 2 2 2 2 2x = x =a. x a -a sin a cos cos)i cn: x = t = ; ( ( -2 2 x = t = ; ( ( -2 2 Lu : V; ', ' ; (( (( - - cost >02 2 2 2t ' '' 't = = 2 2 2 2 2 2 2.acost a cost a - x dx a -a sin dt dt, h bc cos2t. ' '' 't = = 2 2 2 2 2a.cost dx dthay dta - x a -a sin

n y, cng thc nguyn hm khng ph thuc vo bin s nn ta tnh c tch phn theo bin s t mt cch d dng. y ta cn lu : Biu thc trong du tch phn ny l hm s theo bin s t n iu trn [;]. Ta m rng tch phn dng trn nh sau: b) Khi gp dng 2 22 2dxa -u (x)dxhaya -u (x)(a > 0) t .sint . u(x)=a u'(x) dx =a.cost.dt ,; ( ( -2 2tCHUYN :CC PHNG PHP TNH TCH PHN i cn: x = t = ; ( ( -2 2 x = t = ; ( ( -2 2 VD6: Tnh tch phn sau:I62+222= -x +4x -1 dx . Ta c:( ) I62+222= 3 - x -2dxt x - 2 = sint dx = cost.dt 3 3 ,; ( ( -2 2ti cn: 2x = 2+ sint = t =462 2 0 x = 2 sint =0 t =( )I | || |

||\ \ 4 42 20 04400. === 3 - 3sin t 3cost.dt 3cos t.dt3 3 1 3 11+cos2t .dt = t + sin2t = +2 2 2 2 4 2 VD7: Tnh tch phn sau: 220dxI = dx2+x Nhnxt:Tathytamthcbchaimusvnghimnntakhngsdng phngphphsbtnhnhvd4.10vkhngphntchbiuthctrongdutch phn c nh ch 2 v ch 3. t: ( )2x =2tgt dx =2. 1+tg t dt , ; | | |\ t -2 2 i cn: x = 2 2tgt = 2 t =4 x =0 2tgt =0 t =0( )I 24 4420 00= = =2. 1+tg t dt2 2 2dt = t2+2tg t 2 2 8 c) Khi gp dng 2 2dx

a +x(a > 0) Nhnxt:a2+x2=0vnghimnntakhngphntchbiuthctrongdutch phn c nh ch 2 v ch 3. t( )2x =a.tgt dx =a. 1+tg t dt, ; | | |\ t -2 2 CHUYN :CC PHNG PHP TNH TCH PHN i cn: x = t = ; | | |\ -2 2 x = t = ; | | |\ -2 2 Ta xt v d tng t tip theo: VD8: Tnh tch phn sau:I1+221dx=x -2x+3 Nhnxt:Tathytamthcbchaimusvnghimnntaphntchmus c thnh: a2 + u2(x). Ta c: ( )I 1+2 1+22 21 1=dx dx=x -2x+32+ x-1 t ( )22tgt x -1= dx =2. 1+tg t dt , ; | | |\ t -2 2 i cn: x = 1+ tgt =1 t =420 x = 1 tgt =0 t =( )I = 24 4420 00= =2. 1+tg t dt2 2 2dt = t2+2tg t 2 2 8 Vy: d) Khi gp dng ( ) 2 2dx

a +u x(a > 0) Vi tam thc bc hai( )2 2a +u xv nghim tht ( )2u(x)=a.tgt u'(x)dx =a. 1+tg t dt , ; | | |\ t -2 2 i cn: x = t = ; | | |\ -2 2 x = t = ; | | |\ -2 2 Tm li: Phng php i bin s dng 1: nh l: Nu 1. Hm s x = u(t) c o hm lin tc, n iu trn on [;]. 2. Hm s hp f [u(t)] c xc nh trn on [;]. CHUYN :CC PHNG PHP TNH TCH PHN 3. u() = a, u() = b. th [ ]= baf(x) f u(t) u'(t). dx dt T ta rt ra quy tc i bin s dng 1 nh sau: B1: tx = u(t)(vi u(t) l hm c o hm lin tc trn [ ; ] , f(u(t)) xc nh trn [ ; ] v = = ( ) , ( ) u a u b) v xc nh ,B2: Thay vo ta c:( ) I ba= f(u(t)).u'(t)dt = g(t)dt =G(t) =G( ) -GMt s dng khc thng dng phng php i bin s dang 1: * Hm s trong du tch phn cha 2 2 22 2 21a -b xa -b xhayta thng t ax = sintb * Hm s trong du tch phn cha 2 2 22 2 2b x - ab x - a1hayta thng t ax =bsint * Hm s trong du tch phn cha 2 2 21a+b x ta thng t ax = tgtb * Hm s trong du tch phn chax(a -bx)ta thng t 2ax = sin tb BI TP NGH 2: Tnh cc tch phn sau: 1) I120=x 1- x dx 2) I2 120x= dx4- 3x 3) I120x= dx3 +2x - x 4) I221x - 1= dxx

5) I321x +1= dxx(2 - x)6) I120dx=x +x +1 Hng dn:Cu 4: t 1x =sintCu 5: t2x =2sin tVD9: Chng minh rng: Nu hm s f(x) lin tc trn0;( ( 2 th( ) ( ) = 2 20 0f sinx dx f cosx dxp dng phng php trn tnh cc tch phn sau : 1)I424 40sin x= dxsin x +cos x2)I

40= ln(1+tgx)dxCHUYN :CC PHNG PHP TNH TCH PHN Gii VT = ( )20f sinx dxt x = - t dx =-dt2.i cn x =0 t =; x = t =02 2 ( ) VT VP | | | | = = = ||\ \ 0202f sin dt f cosx dx2t (pcm) p dng phng php trn tnh cc tch phn sau : 1)I424 40sin x= dxsin x +cos x t x = - t dx =-dt2.i cn x =0 t =; x = t =02 2 I 44 4 02 24 4 4 44 40 02sin ( - t)cos t cos x2=- dt = dt = dxsin t +cos t sin x +cos xsin ( - t)+cos ( - t)2 2 4 42 2 24 4 4 40 0 0sin x cos x2I = dx + dx = dx = I =2 4 sin x +cos x sin x +cos x. 2)I

40= ln(1+tgx)dxt x = - t dx =-dt4

i cn x =0 t =; x = t =04 4 II 4 4 4 00 0 041-tgt=- ln[1+tg( -t)]dt =ln(1+ )dt =[ln2-ln(1+tgt)]dt =ln2. dt - I4 1+tgtln2 .ln22 = I =4 8 BI TP NGH 3: Tnh cc tch phn sau: CHUYN :CC PHNG PHP TNH TCH PHN

1) 2 2n n0 0sin xdx = cos xdxHD: t x = - t2. 2) Cho a-aI = f(x)dx . CMR: a) Ia0= 2 f(x)dx nu f(x) l hm s chn. b)I = 0 nu f(x) l hm s l.3) Chng minh rng: Nu f(x) l hm s chn th b bx-b 0f(x)dx = f(x)dxa+1.p dng: Tnh 2 2x-22x +1I = dx2+1. 4) Chng minh rng: 0 0xf(sinx)dx = f(sinx)dx2 (HD: t x = - t ) p dng: Tnh 20xsinxI = dx4+sin x. BI TP NGH 4: Tnh cc tch phn sau: (Cc tuyn sinh i hc) a)I =22 220xdx1- x (H TCKT 1997) ( )b) I =13201- x dx(H Y HP 2000) c) I =22 20x 4- x dx(H T.Li 1997)d) I =a2 2 20x a- x dx(H SPHN 2000)e) I =32212dxx 1- x(H TCKT 2000)f) I =14 20dxx+4x+3 (H T.Li 2000) ( )g) I =122-1dx1+x(H N.Ng 2001)h) I =2223dxx x -1(H BKHN 1995) II.4.2. Phng php i bin s loi 2: (Dng nghch) Nu tch phn c dng ( baf u(x) u'(x)dxt: u = u(x) du = u'(x)dxi cn: 2x = b u= u(b)1 x = a u = u(a) ( ) I 21uu= f u du CHUYN :CC PHNG PHP TNH TCH PHN a) Mt s dng c bn thng gp khi i bin s loi 2:(Dng nghch)Trongmtstrnghptnh tch phn bng phng phpphn tch haytnh tch phn bng tch phn i bin s loi 1 khng c nhng ta thy biu thc trong du tch phn c cha: 1. Ly tha th ta th t u bng biu thc bn trong ca biu thc c cha ly tha cao nht. 2. Cn thc th ta th t u bng cn thc. 3. Phn s th ta th t u bng mu s. 4. cosx.dx th ta th t u = sinx. 5. sinx.dx th ta th t u = cosx. 6. 2dxcos x hay (1 + tg2x)dx th ta th t u = tgx. 7. 2dxsin x hay (1 + cotg2x)dx th ta th t u = cotgx. 8. dxx v cha lnx th ta th t u = lnx. VD 10: Tnh cc tch phn sau: 1.a)I13 5 20=(x +1) x dx t: 3 2 2duu=x +1 du=3x dx x dx =3 i cn:x01 u12 I 2 2 2 6 6 65 51 1 1= = = - =du 1 u 2 1 7=u u du3 3 18 18 18 2 b) I

230= (1+sinx ) .cosx.dx (Tng t) 2. a) I220= 4+3x .12x.dxt:2 2 2u= 4+3x u=4+3xCHUYN :CC PHNG PHP TNH TCH PHN 2udu=6xdx 12xdx = 4udui cn:x02 u24 I 4 4 4 3 3 322 2 2= = - =4u 4.4 4.2 224=u.4u.du=4u .du3 3 3 3 b)I22 30= 1+2x .x .dx (HD:I22 20=x . 1+2x .xdx ) t 22 2 2 2-= u 1u=1+2x u=1+2x x2 udu2udu = 4xdx xdx =2 ... c)I1 23 30x= dx1+7xt3 3 3 3 3= = u 1+7x u 1+7x 22 2 2u du3u du=21x dx x dx =7 i cn:x01 u12 2 2 2 2 2 2 21 1 1= = = - =u 1 1u 2 1 3I = du udu7u 7 14 1414 14 3.a)I1 320+x= dxx 1Ta c:I1 220.+x x= dxx 1 t2 2= + = - u x 1 x u 1 = = dudu 2xdx xdx2 i cn:x01 u12 ( ) ( ) ( ) I| | |\ 2 2 21 1 1= = = =u-1 1 1 1 1= du 1- du u-ln|u | 2-ln2-1 1-ln22u 2 u 2 2 b)I2 231=xdxx +2(HD: t 3u=x +2 ) CHUYN :CC PHNG PHP TNH TCH PHN 4.a) I640=sin x.cosx.dx t: u=sinx du=cosx.dxi cn:x0 6 u0 12

I| | |\ 115 22400= =u 1=u du5 160 b)I20sinx= dx1+3cosx(HD: t u=1+3cosx ) c)I20= 1+3sinx.cosxdx (HD: t u=1+3sinx ) 5.a)I20sin2x +sinx= dx1+3cosx ( H khi A 2005) Ta c ( )I 2 20 0sinx 2cosx +1 2sinxcosx +sinx= dx = dx1+3cosx 1+3cosx t 22- u 1u=1+3cosx u=1+3cosx cosx =3 -2udu2udu=-3sinxdx sinxdx =3 i cn:x0 2 u2 1 ( )| || | | |\ \ | | | | ||\ \ 21 222 12 3 3 31-+= + = + - =u 1 -2udu2+13 3 2I = dx = 2u 1 duu 92 2u 2 2.2 2.1 34u 2- 19 3 9 3 3 27 CHUYN :CC PHNG PHP TNH TCH PHN Nhnxt:ivinhngbichacnthc,hcsinhcthtubngbiuthc trongducn,nhngsaukhiibinthtchphnmivncnchacnthcnnvic tnh tip theo s phc tp hn (tc l hc sinh phi a v x). V d: Cch 2 ca cu 5 5.a)I20sin2x +sinx= dx1+3cosx ( H khi A 2005) Ta c ( )I 2 20 0sinx 2cosx +12sinxcosx +sinx= dx = dx1+3cosx 1+3cosx t- u 1u=1+3cosx cosx =3 -dudu=-3sinxdx sinxdx =3 i cn:x0 2 u4 1 ( )4 4 1 12 21 1u u| | | | | |\ \ | || | | | + = |||\ \ \ | | = |\ 4 14 141-1= 2 + = 2 u u +2 u= +4- - 2u 1 -du2+12u+1 1 3 3I = du= du9 u u1 1 1 4u9 9 9 3 u132 4 349 3 3 27 Nhn xt: R rng cch gii 2 t u bng biu thc trong cn thy phc tp hn so vi cch 1. b)I20sin2x.cosx= dx1+cosx (H khi B 2005) 6.a) ( )I= 2420tgx+1dxcos xt:2dxu=tgx+1 du=cos x i cn:x0 4 u1 2 I| | |\ 2 2 321 1= = - =u 8 1 7=u du3 3 3 3 CHUYN :CC PHNG PHP TNH TCH PHN b) I4220tg x - 3tgx +1= dxcos x(HD: t u=tgx ) 7.a) Icotgx 224edxsin x= t:2-dxu=cotgx du=sin x i cn:x 4 2 u 1 0 I 0 1 1u u u0 1 0= = = - =- e du e du e e 1 b) I22p43cotgx +1= dxsin x(HD: t u=3cotgx+1 ) 8.a)I3e11+lnx.dx=xt2u=1+lnx u =1+lnx dx2udu=x i cn:x1 3eu12 I 2 2 2 3 3 321 1 122 = =3u 2.2 2.1 14=u.2udu= u du - =3 3 3 b)I7e31lnx. 1+lnx= dxx t 3 3 3- u=1+lnx u =1+lnx u 1=lnx 2dx3u du=x i cn:x1 7eu12 ( ) ( )I| | | | ||\ \ 2 2 2 7 4 7 43 2 6 31 1 1300. =3 - =3 -7 4 7 4 7u u 2 2= u -1u.3u du=3 u -u du =BI TP NGH 5:1. Tnh cc tch phn sau: ( ) a) I2330= 5sinx -1 cos x.dx

b)I22 30= 1+2x.x .dx c)I1 23 30x= dx1+26x d)Ip20sinx= dx1+3cosx

e)I640=sin x.cosx.dx f)Ip450=cos x.dx g)I62 30=sin x.cos x.dxh)I20= 1+3sinx.cosxdxi)I430= (1+sin2x ) .cos2x.dxj)Ip230= sinx - sin x.dx k)I220sin2x= dx1+cos x 1l)I+4tgx20e= dxcos x 2. Tnh cc tch phn sau: (Cc thi tt nghip) a)I250= sin x.dx (TNTHPT Nm 93-94)b)I2 231x= dxx+2(TNTHPT Nm 95-96) c)I22 31= x+2.x .dx (TNTHPT Nm 96-97) d)I220=cos 4x.dx(TNTHPT Nm 98-99) e)I60= (sin6xsin2x+6).dx(TNTHPT 00-01)f)I 220= (x+sin x)cosx.dx(TNTHPT 04-05) 3. Tnh cc tch phn sau: (Cc thi tuyn sinh i hc) a)I20sin2x +sinx= dx1+3cosx (H khi A 2005) b)I20sin2x.cosx= dx1+cosx (H khi B 2005) ( )c)I2sinx0= e+sinx cosxdx(H khi D 2005) d)I22 20sin2x= dxcos x + 4sin x (H khi A 2006) e)Iln5x -xln3dx=e +2e -3 (H khi B 2006) f)I12x0=(x -2)e dx(H khi D 2006) 4. Tnh cc tch phn sau: (Cc dng khc) a)I1330dx=2x +1b)30= x x+1.dxc)I130dx=1+ x +1 d)Ip302sin2x +3sinx= dx6cosx - 2e)I7e311= dxx 1+lnxf)I3e11+lnx.dx=x.lnx g)I7e31lnx. 1+lnx= dxxh)I4-1ee1= dxx.lnx.ln(lnx)i)I5453x +1= .dxx -1 k)I1x0dx=1+el)Iln5x0= e-1 dx m)Iex0(x +1)= dxx(1+xe )(HD: t = xex) 5. Tnh cc tch phn sau: (Cc thi tuyn sinh i hc) 1) I =7 320x dx1+x(H T.Mi 1997);( )102) I =65 3x 1-x dx(H KTQD 1997) 3) I=3 220sin xdx1+cos x (H QGHN 1997); 4)I10xdx=2x +1(HQGTPHCM 1998) 5) =0cosx sinxdx(HBKHN98); ( )6) I=24 40cos2x sin x+cos x dx (HBKHN 98)7) I =7330x +1dx3x +1(H GTVT 1998); 108) I = xdxe +1(H QGHN 1998) 9) I=30sin xcosxdx(H DLHV 1998);10) I=240sin2xdx1+cos x(HQGTPHCM 1998) ( )11)I=2320sin2x 1+sin x dx(HNT 1999); 12)I=4 24 40sin xdxsin x +cos x (H GTVT 1999) 13)I =12x0dxe +3 (H Con 2000);14)I =ln2 2xx0e dxe +1(H BKHN 2000)15)I=44 40sin4xdxsin x +cos x (H CTh 2000); ( )2116)I = 3dxxx+1 (H NNghip 2000) 017)I=6 26 6sin xdxcos x +sin x (H Hu 2000);18)I=20cosxdxsinx +cosx(HNN1-KB 01) ( )19) I =241dxxx+1 (H Aninh 2001) 20) =220cos xsin2xdx (H NL HCM 2001) 21)I =15 30x 1- x dx(H Lut HCM 2001); 22)I3 78 42x= dx1+x - 2x(CSPNtrang 2002) ( )023)I=23 3cosx - sinx dx (CSPQN 2002); 24)I =4201- 2sin xdx1+sin2x(HC khi B 2003)25)I =2 325dxx x+4(H-C khi A 2003);1026)I =3 2x 1- x dx (H-C khi D 2003) II.5. TCH PHN BNG PHNG PHP TCH PHN TNG PHN:

nh l: Nu u(x) v v(x) l hai hm s c o hm lin tc trn on [a;b] th: [ ]ba= b ba au(x).v'(x) u(x).v(x) v(x).u'(x). dx dx hay [ ]ba= b ba au(x). u(x).v(x) v(x). dv du hay b b ba a a= - u.dv u.v v.du a) Phng php tnh tch phn tng phn: Bc 1: Bin i ( ) ( ) ( )Iba= = b1 2af xdx f x f xdx Bc 2: t ( )( )( )( ) 112 2du =df xu = f xdv = f x dx v = f x dx Bc 3: Tnh Ibbaa= u.v - v.du Ch : Khi tnh tch phn tng phn ta phi nm nguyn tc sau: + Chn php t dv sao cho d xc nh c v + bavdu phi d xc nh hnbaudvb) Mt s dng thng dng phng php tch phn tng phn: Nu biu thc trong du tch phn c cha: Dng 1:( ) ( ) ( ) ( ) ; ; ; nx nxP x sin(nx).dx P x cos(nx).dx P x .edx P x .adxta nn t:nx nxu = P(x)dv = sin(nx)dx hay cos(nx)dx hay e dx hay a dx

Dng 2:( ) ( ) ;aPxlnx.dx Pxlog x.dxta nn t:au =lnx hay u =log xdv = P(x)dx Dng3:hay x xa sin(nx)dx e cos(nx)dx hayhay x xa cos(nx)dx a cos(nx)dx th phi s dng tch phn tng phn n hai ln. (v l mt nguyn hm ca f2(x) ) VD 11: Tnh cc tch phn sau: 1.I =30(3x -1)cos3xdxt: du = 3dxu=3x -11dv =cos3xdx v =sin3x3 I 33 30 00= -2 1 1(3x -1)sin3x sin3xdx =0+ cos3x =-3 33 2. I10=(2x+1)ln(x+1)dxt: 2dxdu =u=ln(x+1)x + 1dv =(2x+1)dxv = x+ x = x(x + 1) I = 11 21200 0- =x(x +x)ln(x+1) xdx 2ln2-21 1=2ln2- =- +ln42 2 3. ( )I12 2x0= 4x - 2x -1 e dx (H GTVT 2004) t: 22x 2xe4x - 2x -11e dx2du = (8x - 2)dxu=v =dv = A - I = 1 12 2x 2x0 01 14x - 2x -1 e - (4x - 1) e dx =2 2( ). A = + =12 2x 201 1 14x - 2x -1 e e2 2 2( ). =12x0(4x - 1)e dxt: 2x1 2xe24x -1e dxdu = 4dxu=v =dv = ( )1 1 10 0 0 = + = +2x 2x 2 2x 21 3 1 1 34x -1 e 2e dx e -e e2 2 2 2 2 A - = -1 I = Nhnxt:Vdtrnldng1catchphntngphn( )nxP x .edx do hng hc sinht u = P(x) nhng do P(x) l tam thc bc hai nn ta tnh tch phn tng phn hai ln. T rt ra nhn xt chung cho hc sinh: Nu P(x) l a thc bc k th tnh tch phn tng phn k ln. 4. I = 4x 204e cos xdx Nhnxt:Dng3catchphntngphnltchphncdng xe sin(nx)dx nhng biu thc trong du tch phn ca v d trn cha 2cos xdo h bc ta s a tch phn v ng dng 3.( ) ( ) I = I I + 4 4 4 4 4x 2 x x x x1 20 0 0 0 04e 2e 2 2e 2 2e 2e 2 cos xdx= 1+cos x dx= 1+cos x dx= dx+ cos x.dx=Ta c:0I44x x 4102e 2e 2e-2 = dx= = I=4x202e 2 cos x.dx t: xx2e dxu =cos x du = -2.sin2xdxdv =2 v = 2e 2-+ I =+ 14x x002e 2 4e sin2xdx =2 cos x =1x04e sin2xdx t: xx2e dx eu=sin x du = 2.cos2xdxdv = 4 v = 4 2B I = 14x x 4004e 2 8e cos2xdx = 4e 4 sin x 2 22 2-2 +B -+II -+I -+ I = | | = = | |\ 44 4=2 4e 415 2 4e 2 4e5 + 2 -+I I I | |= = | |\ 4 4 41 21 14 12e-2+ 2 4e e5 5 5= Nhn xt: v d trn hc sinh phi tnh tch phn tng phn hai ln, trong khi tnh ln hai biuthcxuthintch phnI cntnh ban unntacn gi dng trn l tch phn tng phn lp.Trong dng bitpnykhi lmhc sinh cn lu v du khi s dng cng thc tch phn tng phn. 5. A = 420xdxcos x. T suy ra: B = 420x.tg xdx (H NN Khi B 2000) t 2u=xdu=dxdxv =tgx dv =cos x

4400A - =x.tgx tgxdx = 40d(cosx)+4 cosx =40+ln cosx4 = 1- ln24 2 4 4220 0B=1x.tg xdx =x.( -1)dxcos x= 4 420 0x1x. dx - xdcos x= 21- ln2-4 2 32 6. ( )I322= lnx - x dx (HC Khi D 2004) t:( ) 22x -1(2x - 1)dx (2x - 1)dxdu == u=ln(x-x)x- x xdv =dxv = x - 1 (nguyn hm v = x + c nn thay c = -1 kh mu s) I = 3 322 22x - 1- dx = +1= +x(x -1).ln(x- x) 2ln6 -2ln2 2ln3 1 Nhn xt: Trong dng bi tp tch phn tng phn c cha ln(u(x)) thng xut hin phn s nn rn luyn cho hc sinh kho lo kt hp thm tnh cht ca nguyn hm f(x)dx =F(x)+C vi C l mt hng s thch hp ta c th n gin c phn s cho bc tnh tch phn tip theo n gin hn.Mt v d tng t:I43= 2xln(x - 2)dx7. I dx | | |\ 3230= sin x(H KTrc HN 2001);Nhn xt: v d trn hc sinh phi nhn xt c rng bc u phi i bin s. tu = 3 2 3x u= x 3u=dxi cn:x0 32 | | |\

u02 I du220= 3u sinu I dx220= 3x sinxta bin i nh trn hc sinh d nhn dng tch phn tng phn dng 1. Nhn xt: n y tch phn tip theo c dng 1 ca tch phn tng phn.Do a thc l bc hai nn tnh I, hc sinh phi tnh tch phn tng phn 2 ln:t 2du=6xdx u=3xv =sinx dv =cosx.dx 2103I I4dx = 2220= 6xsinx 3x sinx 1I dx=206xsinxt u=6x du=6dxdv =sinxdx v =-cosx 10 0I 3 dx = + = =22 206cosx 6x.cosx 6x.sinx 2 213 3I I 34 4 = + = Nhnxt:Quavdtrn,tnhtchphnikhihcsinhphipdngchai phng php i bin s loi 2 v tch phn tng phn. V d tng t: (phi hp hai phng php) a)I dx240= sin x b) 1I dx20= x.ln(1+x) c)Iedx240cos lnx=xd) 2I dxcosx0=e sin2x. e) I dxx324ln tgx=cosf) 4I dxx0=e BI TP NGH 6:1. Tnh cc tch phn sau:a)Iln2-x0= xedxb)I60=(12x - 2)cos2xdx c)I620=(2x -4)sin2xdxd)I10=(2x -1)ln(x +1)dx e)I32=(2x -1)ln(x -1)dx f)I224xdx=sin x g)I120= 2xln (x +1)dx h)I2x0=(12x-4+e )sinxdx i)I322=2xln (x-1)dx j)I220=(x +sin x)cosxdx (TNTHPT 2005) 2. Tnh cc tch phn sau: (Cc thi tuyn sinh i hc) a)I43x0=e sin4xdx(H A.Ninh 1997)( ) b)I12x0= x -1 edx(H DLNN-T.Hc 1997) c)I20= x sinxdx(H A.Ninh 1998)d)I | | |\ 240= cos xdx(H DLNN-T.Hc 1998) 21e)I 2lnx= dxx (H Hu 1998) ( )f)I420= x2cos x -1 dx(H TCKT 1998) ( )g)I221lnx +1= dxx (H Con 2000) h)I1021= xlg xdx (H Y Dc 2001) i)I dx | | |\ 3230= sin x(H KTrc HN 2001);j)Ie2 21= x ln xdx(H KT HDng 2002) 1k)Ie2x +1= lnxdxx (HC D b 2-2003); ( )l)I02x 3-1=x e +x+1 dx(HC D.b 2003) m)I2 13 x0= x e dx (HC D b 2-2003); ( )n)I12 -x0= x+2x e dx(H GTVT 2003) III. Kim tra kt qu ca mt bi gii tnh tch phn bng my tnh CASIO fx570-MS Trongmtstrnghpmtsbitchphnphctpgiicktqu nhng cha nh gi c chnh xc ca kt qu l ng hay sai, khi ta c th s dngmytnh cmtayCASIO fx-570MS kim tra kt qu. V d vi thi Khi A nm 2005I20sin2x +sinx= dx1+3cosx ta s dng my tnh nh sau:+ Vi kt qa gii tay l 3427 ta chuyn sang s thp phn 1,259259 + i vi bi tch phn lng gic trc ht chuyn sang ch Rad. + Quy trnh bm my CASIO fx-570MS nh sau: V kt qa my tnh l 1,2593. So vi kt qu gn ng trn ng ngha vi p s bi gii bng tay trn ng. BI TP NGH 7: CU HI TRC NGHIM TCH PHN Cu 1: 102x +1 dxc gi tr bng: A. 2B. 0C. -2D. 3 Cu 2: e20x -1 dxc gi tr bng: A. 1B. 0C. -1D. 12 Cu 3: Chn mnh ng: A. 3424dx4 3 - 2sin x 2B. 3424dx03 - 2sin x 2 C. 3424dx03 - 2sin x 4D. 34241 dx4 3 - 2sin x 2 Cu 4: e1lnx dxx c gi tr bng: A. 1B. 0C. -1D. e Cu 5:( )140x + 2dxc gi tr bng: dx (ALPHA X(sin(2) + sinALPHA X 31 + (0cos ALPHA X ,, )) 2) =SHIFTA. 2115B. 211C. 201D. 2015 Cu 6: 2sinx0e cosx dxc gi tr bng: A. e - 1B. 0C. eD. 1 - e Cu 7: 203 1 +3cosx. sinx dxc gi tr bng: A. 3B. 53C. 1D. 2 Cu 8: 120dx x+x +1 c gi tr bng: A. 3 9B. 9C. 9 3D. 3 3 Cu 9: ( )2212x -1 dx x - x -1 c gi tr bng: A. 2ln3B. 3ln2C. 4ln9D. 9ln4 Cu 10: ( )1204x +2 dx x+x +1 c gi tr bng: A. 3ln2B. 2ln3C. ln4D. ln6 Cu 11: 12-1dx x+2x +2 c gi tr bng: A. ( )ln 2+5 B. ( )ln 2 +5 C. ( )ln 2 +5 D. ( )ln 5 - 2Cu 11: 221dx -3x+6x +1 c gi tr bng: A. 3 3B. 3 9C. 3 12D. 3 15 Cu 12: ( )2214x +6dx x - 2x +3 c gi tr bng: A. ( )4ln 2+3 B. ( )6ln 2+3 C. ( )8ln 2+3 D. ( )10ln 2+3Cu 13:x2 220x+1 dx c gi tr bng: A. 263B. 283C. 323D. 343 Cu 14: x 622dx x -3 c gi tr bng: A. 32B. 36C. 312D. 336 Cu 15: 120dx x+1 c gi tr bng: A. ln 2 B. ln2 C. ( )ln 2 +1 D. ( )ln 2 +2Cu 16: 21dx cosx +1 c gi tr bng: A. 0B. 1C. 2D. 3 Cu 17: 0dx sinx +1 c gi tr bng: A. 0B. 1C. 2D. 3 Cu 18: 0dx sinx - 2cosx - 2c gi tr bng: A.-ln2 B. ln2 C. 1-ln2 D. 1+ln2Cu 19: | | |\ 20sinx -cosxdx sinx +cosx c gi tr bng: A. 1+4B. -1+4C. 1-4D. -1-4 Cu 20: 20cosxdx 11-7sinx -cos x c gi tr bng: A. 1 5- ln3 8B. 1- ln53C. 1 8ln3 5D. 1 5ln3 8 Cu 21: 22-2x +cosxdx 4- sin x c gi tr bng: A. 1ln38B. 1ln36C. 1ln34D. 1ln32 Cu 22: | | |\ 201+sinxln dx 1+cosx c gi tr bng: A. 2B. 32C. 0D. 1 Cu 23: 44 40sin4xdx sin x +cos x c gi tr bng: A.-ln2 B. -ln2 C. -ln3 D. -ln3Cu 24: Cho hm s f(x) lin tc trn R v tha f(-x) + f(x) = cos7x. -2-2f(x)dx c gi tr bng: A. 1635B. 3235C. 2435D. 1235 Cu 25: Cho hm s f(x) lin tc trn R v tha 3 f(-x) + f(x) = cos4x.sin5x . -2-2f(x)dx c gi tr bng: A. 1-4B. 1-2C. 0D. 14 Cu 26: 220x - x dx c gi tr bng: A. 0B. 1C. 2D. 3 Cu 27: 23 2-1x - 2x - x +2 dx c gi tr bng: A. 94B. 3712C. 14D. 4112 Cu 28: 22-3x -3x +2 dx c gi tr bng: A. 592B. 259C. 59-2D. 2-59 Cu 29:x 2205 - 4cos - 4sinx dxc gi tr bng:x | | | | |\ 2 220 05 - 4cos - 4sinx dx= 2sinx -1 dxA. -2 3 - 2 -6B. 2 3 - 2 -6C. 2 3 +2 -6D. 2 3 +2+6 Cu 30: 202cosx -1 dxc gi tr bng:A. 2 3 - 2+3B. 2 3 - 2 -3C. 2 3 - 2+6D. 2 3 - 2 -6 Cu 31: ( )2x-12- 4dx c gi tr bng: A. 12+ln2B. 13+ln2C. 14+ln2D. 15+ln2 Cu 32: 2-1dx 1+ 1- x c gi tr bng: A. ln2 B. 2ln2 C. 3ln2 D.4ln2Cu 33: ( )2-1x - x -1dx c gi tr bng: A. 0B. 1C. 2D. 3 Cu 34: ( )201- x - 1+x dx c gi tr bng: A. 5B. 7C. 9D. 11 Cu 35: 10xlnxdx c gi tr bng: A. 2e +12B. 2e +14C. 2e+11D. 2e +13 Cu 36: 20xcosxdx c gi tr bng: A. +22B. - 22C. +12D. -12 Cu 37: 1x0xe dx c gi tr bng: A. 7B. 5C. 3D. 1 Cu 38: 2x0e sin2x dx c gi tr bng: A.e| | |\ 22- +15B.e| | |\ 21- +15C.e| | |\ 22+15D.e| | |\ 21+15 Cu 39: 22x0e cosx dx c gi tr bng: A. ( )e1+25B. ( )e1- 25C. ( )e12 +15D. ( )e12 -15 Cu 40:( )12x0e x - 2dx c gi tr bng: A. 25 -3e4B. 23e-54C. 23e-52D. 25 -3e2 Cu 41:( )xe0cos lnx dx c gi tr bng: A. ( )e1+12B. ( )e1+12C. ( )e1-12D. ( )e1- +12 Cu 42:( )e0sin lnx dx c gi tr bng: A. ( ) sin1-cos1 e+12 B. ( ) sin1-cos1 e -12C. ( ) cos1- sin1 e+12D. ( ) cos1-sin1 e+12 Cu 43: ex01+sinxe dx 1+cosx c gi tr bng: A.e2B.eC.e 32D.e 2 Cu 44: ( )e2x201+xe dx 1+x c gi tr bng: A. 0B. 1C. eD. 2 Cu 45: ( )ex20xe dx 1+x c gi tr bng: A. e - 22B. e+22C. e -12D. e+12 The end