Làm chủ môn toán trong 30 ngày đại số giải tích 12 quyển 2 (tt)

LE HONG DUC (Chu bien) fl6 Hoang Ha - Dao Thi N goc Ha LAMCHiJ TRONG NGAY QUYEN NBA XUAT BAN D,;J HQC QUOC GIA HA N()I NANG LUC • ==========: DTB < 5,0 5,0 < DTB < 6,5 6,5 < DTB < 8,0 DTB > 8,0 On ~P lfi ki~n thuc cfi Moi ban quay lai IQ trinh L9 trinh Test I ClurO'IIC mot Apd\Jllll91riu1 Kiin thfrc + bai t~p b6 sung IQ trlnh Moi ban quay l~i IQ trinh Test Chll'OII& mol KiSn thuc + bai tijp b6 sung I~ trinh Moi ban quay l~ lq trinh ldt qui< Lq trinh Test K~t qui ~8 Ap ditag 19 criall MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ Djch VI} Van h6a Su pham (Phat hanh sach SPBook), theo hop d6ng chuyen nhuong giira Cong ty TNHH Dich V\l Van h6a Sil pham va cac tac gia Le H6ng Due (Chu bien), D6 Hoang Ha, Dao Thi Ng9c Ha Ban quyen sach thUQC v~ Cong ty NHH ' , Bat cu chep nao ' khong,duoc S\1' dong y cua Cong ty , ~ [ch VI} Van b6a SU' pham deu la bat hop phap va vi pham Luat Xuat ban Vist Nam, ' ' ' luat Ban quyen Quoc te, va Cong iroc Berne ve' ban quyen va SO' hiru tri tue V oi sir menh tao nen nguon tai lieu tri thirc, hfru ich dS giup viec hoc ~P tro nen d8 dang hon, SPBOOK luon mong muon duce cong tac cung cac tac ' , ' , gia, cac thay co giao tren ca mroc de co nhieu hon nira nhirng cuon sach hay dUQ'C phat hanh va d€n voi d(k gia n6i chung va cac em hQC sinh n6i rieng tren ca mroc ' ' , , Cac thay co giao, tac gia c6 nhu cau viet, xuat ban sach xin vui long lien h~ voi chung toi qua: Difn thnai: 024 385 00012 - 0988 852 781 Email: suphambooks@gmail.com Website: spbook.vn Website: maloda.vn Facebook: www.facebook.com/Maloda.vn Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ LOIN6IBAU CHU'ONG Ill: NGUYEN HAM, Ti CH PHAN vA UNG Dt)NG BAI 1: Nguyen ham 10 BAI 2: Mc)t s6 phuong phap tim nguyen ham 56 BAI 3: Tich phan 94 BAI 4: M9t s6 phuong phap tinh tich phan BAI 5: (fog dung tich phan de tinh di¢n tich hinh pb~ng BAI 6: (fog dung tich phan dS tinh thS tich v~t thS OAP so - HlfONG DAN - LOI GIAI CHUONG IV: SO PIIU'C BAI 1: se phirc 210 230 245 313 314 BAI 2: Can b~c hai cua s6 phirc va phuong trlnh b~c bai BAI 3: Dang hrong giac cua s6 plnrc va irng dung OAP 122 so - HlfONG DAN - LCH GIAI 340 364 388 Trang5.J Website: maloda.vn Facebook: www.facebook.com/Maloda.vn Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ Chuang Ill - Nguyen ham, tlch phiin wi trng dung BAI4 - MOT s6 PHUONG PHAP TiNH TicH PHAN j [ A KIEN THCJC CAN NHO Phuong phap d6i bi@n s6 Co so cua phirong phap d6i bien s6 la cong thirc sau: b p • a J f[u(x)Ju'(x)dx = J f(u)du, v6i a= u(a) va J3 = u(b) Tu do, chung ta th§y co hai plnrong phap d6i bi€n: Phuong phdp I: £)~ tinh tich phan: b I= J g(x)dx a Ta thuc hi~n cac buoc: Buac I: Chon: + Phan tich g(x)dx = f [u(x)]u'(x)dx = f [u(x)]d[u(x)] + f)~t u = u(x) Buac 2: Thuc men phep d6i c~n: + Vm x = a thi u = u(a) + V6i x = b thl u = u(b) Buac 3: Khi d6: b u(b) a u(a) J g(x)dx = J f(u)du Phuong phap 2: £)€ tinh tich phan: b I= J f(x)dx, v6i gia tbi~t ham s6 f(x) lien tuc tren [a, b] a ta thuc hien theo cac buoc: Buac 1: Chon x = ~t: u = f1 (x) { dv = fi{x)dx => {du v · Bu&c 3: Khi d6: I = UV I b b a - f vdu • Chu y: Khi SU' dung phuong phap tich phan rung phan dS tinh tich phan chung ta c§n tuan thu cac nguyen tic sau: Lua chon phep d~t dv cho v diroc xac djnh m{>t each da dang b f Tich phan vdu duoc xac dinh m9t each d~ dang hem so voi I a Chung ta c!n nhc cac dang CCY ban sau: Dang 1: Tich phan I= Jxci.lnxdx, voi aeR\{-1} d~t u = lnx Dang 2: Tich phan = JP(x)ecixdx (hoac I= JP(x)eQJCdx) v&i P la m9t da thirc thuoc R[X) va aeR" c16 d~t u = P(x) Trai nghiim Ht sinh thai Hoc Website: maloda.vn di 6.0-SPBook Facebook: www.facebook.com/Maloda.vn Trang 123~ Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ Chuang Ill - Nguyen ham, tlch phiin wi trng dung Dang 3: Tich phan I= JP(x)sina.xdx (hojic JP(x)cosa.xdx) voi P la mc}t da thirc thuoc R[X] va a.eR" d6 d~t u = P(x) Dang 4: Tich phan I = feaxcosbxdx (hoac feaxsin(bx)dx vci a, b f d6 d~t u = cos(bx) (ho~c u = sin(bx)) B PHAN LO,.;.I v A PHUONG PHAP GIA.I cAc D,.;.NG TOAN Phuong phap d8i bil11 d{lng Phuong phap Sfr dung ki€n thirc phuong phap cua phin phuong phap d6i bi~n s6 Vi d1.1 l: Tinh tich phan: I rt/3 I= 11/6 d cosx x sin x - sin x + f>~t t = sinx, suy dt = cosx.dx f>6i c~: 1t l 7t fj •x=-=>t=- •x=-=>t=- Taco: cosxdx dt sin? x-5sinx + t2 -5t+ = dt = -(t-2)(t-3) = (~ ~)dt= + t-3 , d,o: tu {A+ B=0 -2A-3B=l Suy ra: ~ t-2 [(A+B)t-2A-3B]dt (t-2)(t-3) {AB=-1 =l (-1- l_) cos xdx = sin x - sin x + t- t- dt Khid6: I= J"r(-1- 112 Lrrang t-3 124 Website: maloda.vn _l_)dt = t-2 lnl t-311.[J,2 t-2 1/2 = ln 3(6-.fi) 5(4-.fi) Trai nghi/m Hi sinh thai H()c di 6.0-SPBook Facebook: www.facebook.com/Maloda.vn Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ Sung Cac em hQC sinh hay: l(lO: + Tao mi c~: • x=O=>t= 7t • x= - => t=2 Tac6: cosx.sin3 xdx = sin2x.cosx.sinxdx l+sin2x I+ sin x = (t-l)dt 2t = _!_(t -!)dt t Khi d6: I= - f Cl )dt t 1 =-(t-lnltl)I 2 = -(l-ln2) Sang too: Cac em hQC sinh hay: + Tao mot bai toan nrong tu + Tao •vidi} 3: cau hoi trllc nghiern tu n(>i dung vi du tren dx J 2sinsmx2+x.cos' x rt/6 Tinh tich phan I= • D~t t = 2sin2x + cos-x, suy dt = sin2xdx J:'\,t " uo1 can: • x=O=>t=l 7t x=-=>t=- Khid6: 514 I= dt J-t = In It S/4 = In4 l Sang tqo: Cac em hQC sinh hay: + Tao m 0 ~ Bai 88: Tinh cac tich phan sau: a "'s121+ sin 2x + cos2x sin X + cos X n/6 n/3 sm x dx b • rt/6 cos X d J x ~ Bai 89: Tinh cac tich phan sau: a dx n/3 2,r J JI+ sin xdx b J 1t rt/6 sin x sm( x + - ~ Bai 90: Cho ham s6 h(x) = AB a T 1m , d;er., h( x) = ) sin 2x (2 + sin x)2 Acosx + Bcosx (2 + sin x) 2 + sin x b Tinh I= J h(x)dx -ff/2 ~ Bai 91: Tinh cac tich phan sau: a n/J2sin x + cos x + dx b 4sinx+3cosx+5 J cospx.cosqxdx 21! O {p= q p.tq ~ Bai 92: Tinh cac tich phan sau: a «n dx + cosx J «n dx b.J- sin 2x 1114 ~ Bai 93: Tinh cac tich phan sau: rt/2 a J cos? x.sin b xdx J +sm.cos4x x dx rt/4 0 ~ Bai 94: Tinh cac tich phan sau: ,r./2 a dx J + cos x + sin x b o ~ Bai 95: Tinh tich phan I = l ~ J 3sin n/l SID O Trang 204 Website: maloda.vn J dx sm x.cosx cos! x + c2 sin2 x rt/2 Jb x + cos x dx x + 4cos2 x Trdinghiem Ht sinh thai H9c di 0- SPBook Facebook: www.facebook.com/Maloda.vn Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ b Voi x e [O; l] ta c6 nhan xet: O4~t u = x2 + 4, suy du = 2xdx I>6i can: + V6'i x = thl u = 4, + Voi x = thl u = Sx Tu d6· f' o(x2+4)2 dx = ~Js du = ~-J5 - ! 24u2 ~4 Gach 2: Th1,1c hi~n phep bi~n d&i: Sx I Jo (x2 +4)2 ~ J d.x - J' d(x2 +4) o (x2 +4)2 ~o Ta c6 th6 trlnh bay theo cac each sau: I f x (1-x )6x d.x Cach /: Vi~t lai I duoi dang: I= 3 E>~t t = - x3 suy dt = -3x2dx D6ic~: •x=Othit=l, • x = thl t = Khi d6: I=-.!_ f (1-t)t dt=.!_ j(l-t)t dt=.!_ j(t -t')dt=.! (.!_t7 6 I _.!_ts)r O O =-1- 168 Gach 2: Viat l~i I duoi dang: I I I= Jx3(1-x3)6x2dx = J[1-(l-x3)](1-x3)6x2dx 0 = _.!.J[o-x3)6-(1-x3)7]d(l-x3)= _.! [.!.o-x3)' _.!.(1-x3)8] 3 II I 168 a Ta c6 th~ trlnh bay theo cac each sau: Cacb 1: D~t u = f) ~• rx+f , suy ra: u = x + => 2udu = dx A OlC~: + Voi x = thi u = 1, + V 6'i x = th i u = Ji J I sJ O I Tu d6: '-"x + ldx = u2du l Trang 272 Website: maloda.vn = I.Ji I -u3 = -( 2./2 -1) Trdi nghiem Hi sinh thai H9c di 6.0-SPBook Facebook: www.facebook.com/Maloda.vn Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ Cach' 2: f)~t u = x + 1, suy du= dx J., A Doi can: + Voi x = thi u = 1, +Voi x= thi u=2 32 Tit d6: Jx + ldx = J-,.fudu = -u = -( J2-1) I f O I 3 I Cach 3: Thuc hi~n phep bien d6i: = (x + 1)2d(x + 1) = -(x+ 1)2 l I l f Jx+lctx f I = -(2"2-1) b Ta e6 thS trlnh bay theo cac each sau: suy ra: u2 = x2 + l ~ 2udu = 2xdx ~ udu = xdx Cach 1: D~tu = ~, I:\ l " uo1 can: + V oi x = thi u = l, + v cri x = 13 thl u = Jj Tird6: 2 f x.Jl+x dx f u du = 2 O l = ~ - l I Cacb 2: D~t u = x2 + 1, suy du = 2xdx D6i c~: +V6ix=Othlu=l, -ve: x=J3 thl u=4 Ttr d6: 14 Jj 1!4 f x~dx = -f21 Fudu = -u I Ta e6 thS trinh bay theo cac each sau: suy ra: u2 = x2 + l ~ 2udu = 2xdx ~ udu Cach 1: I>~t u = ~, = xdx f)6i c~n: +Voi x= thi u= 1, +V6ix=F3thlu=2 Tu d6: Ji f 0~ 4x 2 dx = 4f-udu = 4f du= I U Trai nghiem Hi sinh thai Hoc Website: maloda.vn 4ul, =4 I di 6.0-SPBook Facebook: www.facebook.com/Maloda.vn Trang 273 ) Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ Dap s6 - Huong d6n - Lui gia! Gach 2: D~t u = x2 + 1, suy du = 2xdx l, A f) OIC~: + Vm x = thl u = 1, + Vcri x = J3 thi u= Tu d6: /3 du du J Jx4x+ dx = 2J Ju = 4J2-Ju O I = 4-Jul I =4 I Gach 3: Th\TC hi~n phep biSn d6i: s Jo Jx x dx = s d( +1 J x + 1) 4~1 = o 2Jx2 + ,{J = o b Ta c6 th~ trinh bay theo cac each sau: Gach 1: D~t u = Jx2 + 1, suy ra: u2 = x2 + ~ 2udu l, = 2xdx ~t u = - cos3x, suy du = 3sin3x.dx l, A f) OIC~: + V&i x = thi u = 0, lt J Tu• d:6: (l-cos3x)sm3xdx l Trang 274 Website: maloda.vn = -J30 udu I = I' I -u e Trdi nghiem Hi sinh thai H9c di 6.0-SPBook Facebook: www.facebook.com/Maloda.vn Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ Gach 2: Thgc hi~n phep bi~n d6i: fo (1-cos3x)sin3x.dx =-l f (l-cos3x).d(l rtf6 rtf6 1· '6 -cos3x)=-(1-cos3x)2 30 =- b Ta c6 the trinh bay theo cac each sau: Gach 1: B~t u = - 4cost, suy du = 4sint.dt Df>i can: + V6i t = thi u = + V6i t = 1t thi u = 1, .! f rt 59 Tud6: 5(5-4cost)4sint.dt = ! -f u du !9 = 94-l ! = u4 I Gach 2: Th1,1c hien phep bien d6i: " ! ! -f (5-4cost) d(5-4cost)= rt fs(S-4cost)4sint.dt = (5-4cosx)4 ! " = ! 94-1 a Ta co thS trinh bay theo cac each sau: Gach 1: f>~t u tame, suy du = ~ cos x = f)f>i c~: +V&ix=Othlu=O, rt + v &i x = thi u = " Tu d6: f costan/ O dx = x f udu O = i !.u2 2 Ca.ch 2: Thuc hi~n phep bi@n d8i: f tacosn~ dxx f tanx.d(tanx)=-tan2l x ,if4 = O rtf4 O 1"'4 l b Ta c6 the trlnh bay thco cac each sau: Cacn 1: D~t u = sinx, suy du = cosx.dx Df>i c~n: + V &i x = thl u = 0, + Vci x = 1t thi u = I Tu d6: "J'2 cos ~dx + sin x = f1~ 1+u O = ln 11 + u f = ln2 o lo = ln2 Trai nghiem Hi sinh thai Hoc di 6.0-SPBook Website: maloda.vn Facebook: www.facebook.com/Maloda.vn Trang 275 ) Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ Dap s6 - Huong d6n - Lui gia! Cach' 2: f)~t u = + sinx, suy du = cosx.dx D6i c~: + V oi x = th i u = 1, + Vci x = re d6: 1t thl u = J l+cos~sinxdx = J ~du 11/2 O I = I I : = ln2 In u Cach' 3: Thuc hi~n phep bi6n d6i: d «n d(l · ) J 1cos~ x = J +~mx + sin x + sin x 11/2 = ln 11 + sin xii"' = ln2 O O Mitt&• a :D~t t = cosx suy dt = -sinx.dx l· f) or " can: • x = thi t = 1, rt • x = thl t = Khi d6: O r'dt I t3dt 1=-f-=J-=J 1l+t2 I( 01+t2 t ) I t dt=[-t2 1nc1+t2)J l+t2 2 11 o 1 = -1n2 2 b D~t u = tame, suy ra: dx 2 du du = -= (1 + tan x)dx = (1 + t )dx :::> dx = - cos2x l+u2 l• A :D OIC~: +V6ix=Othlu=O, , 1t h' + V cnx=-t1u=- J3 Tud6: 11/6 dx [ cos2x /5 2du - I !~~ 11./5 11 - du I 1- u2 - 111 /5( I 1 ) u -1 u + du I +u2 I = (lnlu-11 -lnlu+ll) l"./5 = -lnI [u+l[' u-1 fl J?,+1 = -ln JJ-1 a Ta c6 thB trlnh bay theo cac each sau: Gach I: D~t u = lnx, suy du = dx l Trang 276 Website: maloda.vn x Trdi nghiem Hi sinh thai H9c di 6.0-SPBook Facebook: www.facebook.com/Maloda.vn Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ f>6i can: +Voi x= thi u=O, + V6i x = thi u = ln3 l ln3 Tu d6: J-(Lnx)2dx = ,x ~1n3 ln3 J u2du = u O 30 - Gach 2: Th1,Ic hi~n phep biSn d6i: 3 J-(ln x)2dx = = J (In x)2d(ln x) IX (In x)3 3 I lrr' b Ta co th~ trinh bay theo cac each sau: Gach I: D~t u = ln(2- x), suy du=-~.z., 2-x ~ f) OlC~: + V 6i x = thi u = ln2, + V6'i x = thl u = J Tir d6: ln(2- x) dx 2-x f udu = - -r - ln2 =- 2l1n2 1n2 Cacn 2: Thuc hi~n phep biSn d6i: iln(2-x) J dx 2-x = , -Jln(2-x).d[ln(2-x)] o I' = In (2-x) o 1n22 - -2- a Ta co th€ trinh bay theo cac each sau: Cacti 1: f>~t u = x3, suy du= 3x2dx f>6i c~: + Voi x = thi u = 1, + V6i x = thi u = Tu de: 18 Jx c">dx = -Jeudu = I I ~us = I -(e8-c) xl Gach 2: f>~t u = e"1, suy du = 3x2 e dx f>6i c~n: + V 6i x = l thi u = e, -t-V6ix=2tbiu=e8 re d6: f x2e" dx = ! J 3e I du = ~1c• = !.(e 3e Gach 3: ThtJC hien phep bi8n d6i: 12 c"J J x2e"1dx = -Jex'd(x3) = I I -e) I = -(e8-e) I Trai nghiem Hi sinh thai Hoc di 6.0-SPBook Website: maloda.vn Facebook: www.facebook.com/Maloda.vn Trang 277 ) Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ Dap s6 - Huong d6n - Lui gia! b Ta c6 th€ trlnh bay theo cac each sau: Cacb 1: DJt u = 3x3, suy du = 9x2dx D6i c~n: + Voi x = thi u = 0, + v cri x = thl u = 3 J u J s Tit d6: x2e3 clx = !._ eudu = 90 90 x) Cacb 2: D~t u = e3x3 , suy du t, = !_(e3 -1) = 9x2 e x' dx A £) OlC~: + v cri x = o thl u = 1, + Voi x = thi u = e' re do: f x e 3•' dx = ! J ~1e' = ! ( du = e3 -1) Cacn 3: Thuc hien phep bi6n d6i: a D~t t = lnx suy dt = clx x ,., f)01 can: A • x = => t = 0, • x==e cc t=s l ) 01 = -.43 f (t2+1)dt = (1-t3+t Khid6:I= b Ta c6 the hra chon mot hai each trinh bay sau: Gach 1: f)~t x = sint, - ~ ~ t ~ ~ r,A• suy dx = cost.dt A :uot can: • x = thi t = 0, Ji • x= - 1t thi t = - Khi d6: I= Tsin2 t.cos tdt Jt-sin2t = -1 l 1114 20 = f (l-cos2t)dt Trang 278 Website: maloda.vn 'Tsin2 t.cos tdt = [cost] O = -1 J O sin2 t.cos tdt cost ( t sm2t ) "'4 - -rt - -1 Trdi nghiem Hi sinh thai H9c di 6.0-SPBook Facebook: www.facebook.com/Maloda.vn Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ Gach 2: D~t x = cost, t e [O; n] suy dx = -sint.dt l, A f) OIC~: • x=Othlt=~ Ji • x= 2' thl t = rt/4 1112 J1- ~ · d d n/4 · 1112 I sin t I : f(x) = x2ex, gia sir: F(x) = fx2ex = (ax2 +bx+ c)ex + C L§y dao ham hai v~ cua (3), ta duce: x2ex = [ax2 + (2a + b)x + b + c)e"' £>6ng nhAt h~ sb, ta duce: { (3) (4) ;: ~ b = ~ {:: ~2 b+c=O c=2 I Khi d6: F(x) = (x2 - 2x + 2)ex + C=> J x2exdx I = (x2 - 2x + 2)ext = e - _ U== a E>~t: { ln X s ~ dv= x dx l du=-dx X x v=6 J J x ln x - -I x s dx = -32ln2 Kh1 d'o: x s In xdx = - l ~ I Trang 282 Website: maloda.vn I I - ~2 x 36 I 32ln2 - -3 Trdinghiem Hi sinh thai H9c di 6.0-SPBook Facebook: www.facebook.com/Maloda.vn Hotline: 0972.853.304 MALODA.VN - KHO SÁCH QUÝ, THI HẾT BÍ b D~t: y {u = cos x {du = +sin xdx • dv=exdx v=ex Khid6: I" + f ex sin xdx = -en It I= excosx O " Jex sin xdx + O (1) O Vci tich phan J, d~t: u = sin x {du= cosxdx => J = e'sinx { dv =e'idx ~ y:;;;:ex I" - O f•ex cos xdx = -1 (2) o Thay (2) vao (1), ta dircc: I= -en - - I~ 21 =-en - ~I= _ _!_(en+ 1) a Sir dung d6ng nhAt thirc: x = ta duce: x (1 + 2x)3 = 2I I'[ (1 + I2x) l) _!_ [ - 1-] (1 + 2x)2 (1 + 2x)3 Khi do: T= ! ( + 2x - - I l (1 + 2x)3 dx = 2I [ I I 2(1 + 2x) + 4(1 + 2x)2 ] I' I 18 b Bi~n d6i: _4_x_+_l_l_ = 4x+ll x2 +5x+6 (x+2)(x+3) =_A_+ x+2 _B_= (A+B)x+3A+2B x+3 (x+2)(x+3) D6ng nh§.t dbg thirc, ta duce: A+B=4 {3A + 2B:;;;: 11 Dod6: I= ¢:> {A=3 => 4x+ll = +-B =I x2 + 5x + x +2 x +3 ) ) dx=(3lnlx+2l+lnlx+31) fI( +x+2 x+3 O : x2 +3x+10 a Bien 0; oi: x2+2x+9 =1+ x+l x2+2x+9 l =ln- o = + _1 2x+2 2.x2+2x+9· Khi d6: I= J (l+-.2l x 2x+ 2x+ 2+ )dx=(x+ 2 -lnlx -2x+91) I I ° I =x+ -ln- Trai nghiem Hi sinh thai Hoc di 6.0-SPBook Website: maloda.vn Facebook: www.facebook.com/Maloda.vn Trang 283 ) Hotline: 0972.853.304 ... + 2, suy ra: 2tdt = e'dx ~ dx = -2t -2 = D6ic~: • x=O=>t=JJ • x = ln2 => t = Khid6: wi trng dung • ~~r = _1_ ln = _1_ In (2- J2)( Jj + J2) 2J2 ~.,13 2J2 (2+ J2)(fi-J2)' I= J2~ J3t2 -2 Sang tpo: Cac... B.ln- 2x.dx ~ sin bang: 2 2sm x+cos x C ln~4 + I D ln + ·1 ' , h dx bL 1c1 tn cua he phAan lnJ2 r; ;:: ang: o ve" + A _I_Jn (2+ ,,J2)(.fi +,,J2) J2 (2- ,,J2)(.fi-J2) C.-1-ln (2- Ji)(JJ +Ji) J2 (2. .. 2- v -1 = t: Ji 12 f 1 12 (1 v3dv - dv = fj v 2) v = Ji /2 f fie-) _)dv "2 v2 ( 1n J t 1v-ll)Ji v v+l 12 v2 - = fj [2- 1 12 fi. +2 J2 + -ln -=-J 3 (2- .fi.) Sang lflO: Cac em hQC sinh hay: + Tao