515.076 GI-103N TRAN TUAN ANH Gliil NHANH BAI TOAN HGIIYJNHAH &TiCH PHAN D A N H C H O H Q C S I H H L0P11-12 TRAN TUAN ANH m r m m m • » - ^ NGUYEN HAM ^ T I C H PHAN T!U/ VIEN TiNHBlNH THUAI>2 NHA XUAT BAN DAI HQC QUOC GIA THANH PHO HO CHI MINH GIAI NHANH BAI TOAN NGUYEN HAM VA TICH PHAN La>i noi dau Nha xua't ban DHQG-HCM va tdc gii/doi tdc lien ke't gifl ban quy^n® Copyright © by VNU-HCM Publishing House and author/co-partnership All rights reserved Viec giai mot bai toan noi chung la mot qua trinh tu cao do, dua tren hilu biet cua nguai giai toan Viec tinh mot bai toan nguyen ham hay mot bai toan tich phan cung vay Co nguai tham chi khong giai dugc, c6 nguai giai dugc nhimg can qua trinh may mo rat lau, thu het each den each khac mai giai xong, c6 nguai lai tim dugc each giai rat nhanh Vay dau la bi quyet de giai nhanh dugc mot bai toan nguyen ham, mot bai toan tich phan noi rieng? Cach ren luyen de c6 each giai nhanh? Cuon sach viet nhSm dem lai cho ban doe nhimg each hieu, nhijng huang di, thu thuat de tilp can nhanh tai lai giai thoa dang cho mot bai toan nguyen ham, mot bai toan tich phan Cac cong thuc dua tai nguai doc khong CONG TY TNHH M^T THANH VIEN SACH VIET 391/15A Hajnh T i n Phat, P.T§n Ttw^n Dong, QuSn 7, TP.HCM, BT: {06} Jf.720.837 • F a x P) 38.726,052 • MST: 03114307135 Email: W i f i a c l R f i e t c o x o m - Website: «»w.sachvie!co.«ni mang tinh ap dat ma theo huang de hieu, de nha de nguai doe c6 thien cam han ve cac cong thuc do, phuc vu cho viec van dung tinh toan sau Cuon sach viet theo loi dien giang nen kho tranh khoi khiem khuyet, rat mong nhan dugc nhung gop y thiet thuc ciia ban doe gan xa Xin chan cam an nhung gop y, chi dan cua quy thay: - TS Nguyen Viet Dong, Truang Bo mon Giao due Toan hoc, DHKHTN, DHQG TP H6 Chi Minh - Thhy Nguyen Dinh Do, Pho Hieu truong Truang THPT Thanh Nhan TP Ho Chi Minh - ThSy Le Hoanh Sir, Giang vien DHQG TP HCM Te Luat i ' - Truang DH Kinh ' - Thhy Nguyen Tat Thu, Giao vien Truang chuyen Luang Th6 Vinh Bien Hoa - Dong Nai Tran Tuan Anh Xufi't ban nam 2013 G I A I NHANH B A I TOAN T I C H PHAN T R O N G D E T H I T U Y E N S I N H D A I H O C NAM Cau 1; Tinh tich phan / = | — ~ 1" • (E>H kh6i A, A i - 2013) Cdch gidi thong thir&ng w - I Cdch 1: / = ——.lnxdx= + A = In \nxdx + In xdx Ta xet: xdx Dat u = lnx=> du = —dx ; dv = dx=> v = x X In xdx = x\nx -1 + h = ' -dx = 2\n2-\ X In xdx Dat u = \nx^ , du = — dx; dv = X -1 rfX = > V = — J X \X f 1\ In xdx = ^f 1, dx = — Inx X X = —Inx X I + X = -ln2— 2 Vay / = / , + / = n - l + - l n - i = - ( n - ) V-1 Dat /• = In X = > X In xdx - 1- = e' va dx = e' dt Inxc/x Doi can: x = l = > / = 0; x = 2=i>/ = ln2 In / 1= , \n Cau 2; Tinh tich phan / = ^x^l-x'dx In ^\\—L-\e'dt= \[e'-e")dt= \td(e'+6-) Cdch giai thong thu&ng V = t(e'+e-) In In • " nen ta chon an phu x = V2 sin? Cdch 1: Do dau hieu '(e'+e-')dt Dat X = y/l sin t => dx^-Jl Doi c a n : x = = ln2 V 2, Cdch 3: cdc ban di y quan he giita X X X ^x'-\^ Xet tich phan J = 272 |sin / cos^ tdt Vdy ta CO the giai dx = d x + - dx = x'-\ ; quan he — va \ : -^dx = d nhanh bdi todn tren nhu sau : 1= / = 0;x = => / = — = 27^ sin ^ cos/ cos/L// = 2V2 sin/cos tdt giiia X vd la : \dx = dx 1- In xdx = Dat u = cos/ ^du = -s'mtdt X x+ - - V \^2 = Inx x + - 1^ In xdx = In xd x + * O T a c o : J = -l4l x+ - X rv 1+ 72 K Doi can : / = =o w = 1;/= — => w = — = ln n n Taco: / = JV2 sin / V2 - sin" /.72 cos /Jr = V2 Jsin cos Vl - sin^ / J/ Cac/i ^w/ nhanh CO : cos tdt, te - i l(51n2-3) Vay / = | ( n - ) Do do, ta ( D H kh6i B - 2013 1^ \u^du = 2^ \u^du = 2^.— 7^ = 272-1 r 02— ( dx = In X x + — = |(51n2-3) X 1' — Vay / = 272-1 Cdch 2: Theo kinh nghiem thi thay can thuc ta dat can thuc la an phu ! Dat / = 72-x^ ^t^ =2-x^ =>tdt = -xdx I,^/ giai that nhanh ggn so v&i hai cdch tren ! D6i can:' x = => / = 72;x = =i> / = Taco: I = -\t^dt= \t^^t= — V2 2V2-1 Cdch 2: Ta c6 : /= i Cdch gidi nhanh Cdch 3: Cdc ban de y quan he giua x vd x^ la: xdx = ~^d{^x~^ = -^d{2-x^y ;—'—dx= X +\ 2x Xet tich phan J = Nen viec ta chon an phu cdch 2) la hodn todn tu nhien ! khong mang tinh dp dat cua kinh nghiem 2x_ \dx+ \— dx^\ J 0r - -4-1 x'+\ J V 2x x'+l -dx •dx x^+1 (0 t ; dx^ x ^ + 1J Dat x = t a n / = > ( i x = —^—dt = {\-^ian-t]dt, cos"/ ^ ' ti '2 suy nghi Id : "thay cd can thiic thi dat can thitc la an phu" Chung ta c6 the gidi nhanh nhie sau: Doican: x = ^^^-p!- / = 0;x = = ^ / = — x42^dx = ^\2-xjd(2-x') = _ 2V2-I 0" Tadugc / = f ^ ^ ( l tan / + sin/ + t a n ^ / ) ^ / = f ^ c J/ = 2xdx D6ican: x = (x^ + ) Nen viec ta chon dn phu t = x' -\-\(o cdch 1) Id hodn todn tu nhien ! x'+\ Dat / = 2xdx = ci(x^ j = -dx / = = X + -flX = I , dx = \dx + f ^"^ dx 0 + ln x ^ + l i/(x'+l) =x x^+1 = l + ln2 = ln2 LM gidi that nhanh gon ! Vay / = l + l n D6 CO each nhin "tudng minh" vh each giai nhanh Nguyen ham va Tich phan, mai ban doc t i m hieu nhirng kien giai cuon sach ! ChLPcng N G U Y E N H A M (2) Cong thii-c : \dx=l Ta suy nghi : ham so nao c6 dao ham bac nhat bang 1? De dang nhan thay la X vi x' — Vay ta c6 cong thuc thu haii Bai NGUYEN H A M \dx = x + C Dinh nghIa (3) Cong thii-c : Cho ham so f(x) xac dinh tren K (K la khoang ho^c doan hoac nua khoang cua M ) Ham s6 F(x) dugc goi la nguyen ham ciia ham s6 f(x) tren K nSu x"dx =? Ta suy nghi: ham s6 nao c6 dao ham bac nhat bang jc"? Chung ta lien tuong toi cong thuc dao ham {x")' = nx"'^ hay F'(x) = f(x) vai mgi x thugc K = x" Ta thay n-\^a Mgi ham s6 f(x) lien tuc tren K d^u c6 nguyen ham tren K hay « = a + , thu dugc cong thuc = X « +l Sau nay, yeu chu tim nguyen ham cua mot ham s6 dugc hieu la tim nguyen = x" Vay la ham so hay ham tren tung khoang xac dinh cua no F(x) la mot nguyen ham ciia ham f(x) thi F(x) + C (C la hang s6) la ho nguyen ham cua ham f(x) hay tich phan hk dinh cua ham f(x) Ki hieu : fix)dx c6 dao ham bac nhat bang x" Suy a +\ a+\ cong thuc thu ba : x"dx=-— + C (a^-1) ar + = F{x) + C (4) Cong thuc : f—c/x =? Ta suy nghi: ham so nao c6 dao ham bac nhat Vi du bang — Ta lien tuong toi cong thuc ( i n x ) =— thi thu duac cong thuc a) J2xdx = x^+C vi ( x ' + C ) ' = 2x b) X X cosxdx = smx + C vi (sinx + C)' = cosx * Luu y: di hiiu nhanh nhung noi dung kien thuc cuon sdch nay, ban doc nen ren luyen thdnh thgo viec tinh dgo ham ! Chiing ta lay dau gia tri tuyet doi vi dieu kien ciia ham Logarit! V (5) Cong thij-c : Tinh chat thii- nhat f'{x)dx=fix) \-dx^\nx+C J a''dx=7 Ta suy nghi : ham so nao c6 dao ham bac nhk bang a''? Tu cong thuc tinh dao ham quen thugc (^a"^ = a ' ' l n a +C Tinh chat thu nhSt dugc suy true tilp tir dinh nghia nguyen ham Trong thuc hanh, tinh chk giup ta tim nguyen ham cua mot ham so don gian, cung vlna; = a", tiic la ham so In a hay c6 dao ham bac nh^t bang a" Vay ta d l dang nhu viec xac dinh lai nguyen ham tim c6 dung khong theo each nghi: ''muon tim nguyen ham ciia ham so f(x), chiing ta tim ham so md dgo ham bgc nhat thu dugc cong thiic cm no phdi chinh la f(x)'\i each hieu do, chung ta c6 the lap Bang a +C a'dx = — (a>0,fl^l) \na (6) Cong thuc : cong thuc nguyen ham co ban nhu sau : e''dx=? Ta suy nghi: ham so nao c6 dao ham bac nhSt J (1) Cong thirc : Qdx =? Ta suy nghi : ham so nao c6 dao ham bac nhat bang 0? Hien nhien la hang so ! Vay ta c6 cong thuc thii nhSt: Qdx = C bang e"') De dang ta nhan thay la ham e' vi (e'') = € ' , suy cong thiic thu sau : e^dx = 6" + C Cong thuc thii sau la truofng hgp rieng ciia cong thiic thii nam thay "a" bang "e" ! (7) Cong thu-c : jcosxdx =? Ta suy nghi : ham so nao c6 dao ham b$c nhk bang cosx? Tir cong thuc quen thuoc (sinx) thuc thu bay la : =cosx, ta c6 cong — sin^ X — cos^ X sin^ X cosxi/x =sinx + C (8) Cong thii'c : sin xdx =? Ta suy nghT: ham so nao c6 dao ham bac nhat sin^ X sm^ X smx ^ Vay ham so c6 dao ham bac nhat bang ^ la sin^ X sin^ X ham so - cos X hay ( - cotx) Suy cong thuc thu m u a i : sinx bang sinx? Tu cong thuc quen thupc (cosx) = - s i n x hay (-cosx) = s i n x , dx = - cot X + C ta CO ham so ma dao ham bac nhat cua no bang "sinx" la " - cosx", suy cong thuc thu tarn la : | sin xdx = - cos x + C (9) Cong thuc : hop tren ! chung ta dir doan ham so can tim c6 dang A cosx + sinx A "cos^x") Ta c6: cos a; ' ^ cos cos ' - Vay ham so c6 dao ham bac cos X , ro rang neu chon A = sinx thi X cos^x + sin^o; X X —Y~ ^ cos x +C •dx = ? Ta suy nghi : ham so nao c6 dao ham < a + b-0 b = -l thi F ( x ) la mot nguyen ham cua f(x) Vay vai a =l va Vi du 6, Chung minh rang F(x) - sin xe"" la mot nguyen ham cua ham so / ( x ) = (sin X + cos x y Gidi + 1)^ dx X + — + -I CO : J 7^ = X X -I - dx -I x = ^ + - e^x^ dx X- + Tap xac djnh cua F(x) va / ( x ) la IR Taco: F ' ( x ) = ( s i n x ) ' ^ ' ' + sinx(e'')' 4^2!^ + In = cos xe"" + sin xe' = (cos x + sin x)e'' = / ( x ) + C X Vay F(x) = sin xe' la mot nguyen ham cua ham so / ( x ) = (sin x + cos x •dx e'x^ 3 ,3 JC JC X X dx = — - + In X X c) T a c o ': 73 = JT.2,'dx + dx x = J(2.3)'dx X ~e'' dx BAITAP Tinh : +C 25a.'' + 2 ' + 1991 = jG'dx d) T a c o : re = \—; d u = d(l + - ) + C • (ta hieu suy nghi " x + " la "u") Quan , —1 va , 1— 1; —, dx = -—1 a, giua , e — d\-—e-1 X nen ta co X X Do do, ta CO thi chon an phu la w = + — X X Dat u = Gidi a) Phdn tick bdi todn : Niu chua dugc biet din quan he giua \ — thi X Taco: i X that khong de de chung ta tim phep dgt an phu! Cdc ban de y quan he = r • 2J / sin —d — X 2^ X i =^ du = d X _ i f sin(2u) du = - i ( - - c o s u ) + C - - c o s u + C9.J ^ 2 Thay u = — ta duac: I = — cos X + C- * Nhan xet: Niu dd thao viec sic dung phuang phdp nay, cdc ban CO thi trinh bay lai gidi nhanh han nhu sau : 1+- -dx = e \ b) = / — X Thay u = 21nx + tadugc: 1+- -1 ^of X X X — 10 / ' / s m — cos — ax = — i s m — c o s —a X "J 1+ =ifuMu=1.^:^+0=^+0 6:1 CO a)Ii = Ta = f sin 2^ X 20 ^ ( I n x + 3)^" ^ 20 ^ b) Phdn tich bai todn: Cdc ban dSy quan he giua — va \nx : —dx = d{\n x) x X —a X l n ^ x + 51n^x , 21n^ x + 51n^ x \ , , ^ , dx = d[\n x) Do vay, ta chon an phu Inx xlnx yi^n ta CO = — cos /fl w = In X Q u a n he g i u a — v a line LM gidi cda bai todn X — dx X = — d{a a \nx + In^ r 2211nn-''xx + l n ' x dx In X Ta CO (In x ) = — nen quan he can xet giiia — va I n x la : X + In^ x Inx d(lnx) Dat u = Inx => du = d(lnx) b) •2u' + u ' Taco: = j : 2u' du u (ta hieu cong thuc tren mot each dan gidn nhu sau : dua — vdo vi + 5u' — du = u J (2u^ + 5u) du X phdn thi {a I n x -\-h), vai a ^Ovdb tiiyy tren R) V i d u Tinh : In X + -dx; b ) I , = / 21n'x + l n ' x xlnx X Thay u = Inx ta duac : i ='^i^RJ^ +^A\}12^+ c • ' * Nhan xet: Niu da thao viec su dung phuang phdp nay, cdc ban CO the trinh bay lai gidi nhanh han nhu sau: dx Gidi: a) , = / i In X + p ^dx= \ K9 /•-(21nx + d(21nx + 3) a) Phan tich bai todn: Cdc ban di y quan he giita — va I n x ; X _(21nx + 3r ^dx = ^d(21na, + 3) nen ta c6 i l l ! l f l L a ! x = l(21njc + 3)V(21nx + 3) vay, ta chon an phu Id u (2 In 2lnx + Ij = J Dat u = p i , X + ^dx In X + , + 3) d(21nx + r In^ x + In^ X , / xlnx dx = r In^ x + In^ x , , / ; d(lnx) Inx ^ "-/, ^2 ^, 1,., X 2(lrix)^ 5(lnx)^ „ ( l n x ) +51nx d(\nx) = -A L _A L 4- Q -•^ ^ J >9 = J ^ ( n x + 3) => du = d ( In X 20 b) I Ldi gidi cda bai todn n Do ^ ^ 3) Q u a n he giira e^ v a ae^ + b Ta CO {^ae' = ae' nen quan he can xet giira va ae' + b la: Cdc ban de y quan he giiea (a^O) a (ta hiiu cong thuc tren mot each dan gicin nhu sau : dua • e'^^dx + • dx — + vao vi -d{e' + +!)• LM gidi cua bdi todn V i d u Tinh : ' e'^dx = d(e* + 1) nen ta c6: Do do, ta chon dn phu la u = e" + \ phdn thi {ae"" + h), voi a ^Ovdb tiiyytren M) [—1—dx b) va e'^ + l + e- X -J 2e' +1 -dx = —-dx = -dx = -.e'dx e^+1 e'+\ 1+- Gidi a)Phan tich bdi todn : Cdc ban de y quan he giua ^ va 2e^ + • + Dat u = e ' ' + l ^ d u = d ( e ' ' + l ) e'^dx = ^ d(2e^ + 1) nen ta c6 dx 2e" + e'dx = 2e" + Taco: I = f-du u - d(2e' + 1) 2e^ + l (i(2e^ +1) Do vgy ta chon dnphu la u = 2e' +\ 2e' +1 Ld'i gidi cua bdi todn I = r-^-Hl-dx= r - ^ e M x = id(2e''+l) J 2e^ + ^ 2e^ + ^ 2e'' + ' = ln|u|+C ; Thay u = e" + ta dugc: * Nhan xet: Neu da thao viec su dung phuang phdp nay, cdc ban CO the trinh bay lai gidi nhanh han nhu sau: a) I = r — d ' -'20^+1 b) I2 = x = r — - — - d ( e ' ' + l ) = -ln(2e^+1) + C = r-.^^d(2e^+l) = - f—^d(2e^+l) Ta CO r l , ' ^ 1 = In u + C tuyet d6i vi 2e'' +1 > 0) l + e-X -dx = ^ -dx e^+1 Ta CO (sinx) =cosx va (cosx) = - s i n x nen quan he can xet giua sin a; X la: s inxdx = — — d(a cos x + b) a (Ta hieu cong thuc tren mot cdch dan gidn nhu sau: dim cos x vao 1+ ^ f — ^ d(e'' + 1) = ln(e^ + 1) + C • c'' + cos xdx = — d{a s i n x + b ) a b) Phdn tich bdi todn : Ta Men doi h = 1+ Quan he giua sinx va cosx va cos 1_ -dx = ^ 26" + Thay u = 26" + ta dugc : = - ln(2e"' + 1) + C • (ta khong lay dau gia tri r 2e^ + ^ -dx = l +e 20" + Dat u = 26" +1 =^ du = d ( e ' + ) = ln(e" + 1) + C -dx = -dx viphdn (asinx+h); dua sinx vao vi phdn -(acosx + b), vai * Nhan x4t: V i du 5.-Tinh : b) = Jcos^xsin'xdx; a) Jcosxe-'"""+'da: Neu da thgo viec sir dung phucmg phdp nay, cdc bgn CO thi trinh bay lai gidi nhanh han nhu sau : a) I j = J cos^ x sin^ x d x = J cos x ( l — sin^ x) sin^ xdx Gidi — J (I- a) Phan tich bai todn : Ta bien doi : cos^ X sin'^ x = cos x cos^ x sin^ x = cos x{\ sin^ x ) sin^ x Cdc ban de y quan he giua sinx va cosx; cosxdx sin^ X = d(sinx) b)/, = LM gidi cua bai todn f{l- / cos xe - = J cos^ x sin^ x d x = J cos x cos^ x siii^ x d x cos x ( l - sin^ x ) sin^ x d x = sin^ x nen ta c6 cosx(l -sin^x)sin^xdx = (1 -sin^x)sin^xd(siwc) Do vdy, ta chon an phu la u = sinx =J sin^ x ) sin^ x d ( s i n x) = J* (sin^ x - sin^ x ) d ( s i n x ) - 33sinx+2 si ^ sm x ) sin^ xd(sin x ) sin + c - 1+2 dx^j -38mx+2 +c Quan he gifra sin^x, cos^x va sin2x Ta CO (sin^ x ) = s i n x c o s x = sin2x Dat u = sinx => du = d(sinx) c / ( - s i n x + 2) va (cos^ x) = - c o s x s i n x = - s i n x nen quan he can xet giua sin^x, cos'^x va sin2x la : Taco: = J (1 - u ' ) u M u = J ( u ' - u ' ) d u = y ™ • i T (sinx)^ Thay u =^ sinjc ta duac: I =-^^ ^ ' - y+0- (sinx)^ „ ^ -^ —+C5 b) Phan tich bai todn : Cdc ban de y quan he giua sinx va cosx; cos xdx = d{-3 sin 2xdx = — d{a sin^ x -\-b) a (ta hieu cdng thuc tren mot cdch don gidn nhu sau: dua sin2x vao vi phan (a sin^ x + b) hogc —[a cos^ x + b), voi a ^0 va b tuyy tren M) V i du Tinh : sin x + 2) nen ta c6 a) cos = — e-'^'^'^^di-?, Do vdy, ta chon an phu Id u = -3sinx sin 2xdx = — — d{a cos^ x + b) a = j(3siu^x+l)sin2x(ix; b) = i n22 xz r _ ^ _ s is n ^ sin x + 2)- V2sin'rz;+• cos rdX • X Gidi + a) Phan tich bai todn : Cdc ban diyquan he giCta sin^x va sin2x; LM gidi cua bai todn = Jcosxe-'''"''^'dx = J ^ e - " ' " ' ^ + ' ( i ( - s i n x + 2) Thay u = - sin x + ta dugc: ta c6 (3sin^ x+l)sin2xda; L&i gidi cua bai todn : +C = — nen = •^(3sin^ x + l ) d ( s i n ^ x + ) Do vdy, ta chon dnphu la u = 3sin^x + I o Dat u = - sin x + =^ du = d ( - sin x + ) Taco: I = — f e M u - — e " ' ^ sin2xrfa; = - d ( s i n ^ x + ) = J(3sin'x+l)sin2xdx e-3»""'+2 ^ = J^(3sin^ x+l)d(3sin^ x+1)- Q E>at u = s i n ^ x + ^ du = d(3sin^ x + ) Taco : I = - / udu = - hC= i hC- Thay u = s i n ' x + l tadugc : I ^ ( s i n x + ) ' • * Nhan xet: - A'ew trinh bay lai gidi nhanh han nhu sau : a) b) Phan tick bai todn : Ta bien doi: sin 2x sin 2x sin 2x V2sin^ X + 3cos^ x -^2(sin^ x + cos^ re) + cos^ x Vs + cos^x thgo viec sir dung phuang phdp nay, cdc ban c6 the = J(Ssin^ s i n 2a: _ V + cos^a; =d{2 + cos^ x)- V s i n ^ X + 3cos^ x = - '(2 + cos^x) t/(2 + cos^x) = -2V(2 + cos^x) + C V2 + cos^x u = V2 + cos" X cfe Z>/ew //zii-c i/j/OT i/aw nguyen ham khong can thuc - A'ew chung ta de y den quan he giua = s\n2x , r —1 \ -/ , dx = J , c^(2 + cos^ x) • ^ V s i n ^ x + 3cos^x ^ V + cos^ x r X => = s i n a; va c o s x thi chung ta c6 them each gidi theo huang khdc nhu sau : a) / j = J(3sin^ L&i gidi cua bdi todn : Dat u = V2 + cos^ yJ2 + cos^ x do, ta c6 the chon an /fl w = + cos^ X /zoac w = V2 + cos^x r/-o«g truang hap ta nen chon h=J x+l)c/(3sin^ x+1) _(3sin^x+l)^ Cdc ban di y quan he giua cos^x va sin2x.- sin2xda; = -d{2 + cos^ x) nen ta CO x + l ) s i n x c ? x = j^(3sin^ X x4-l)(ix x + l ) r f ( s i n x ) = y (6sin^ x + s i n x ) ( i ( s i n x ) J2sinx(3sin^ „ sin^ = + cos^ x => du^ = d(2 + cos^ x ) x + l ) s i n x ( i x = j2sinxcosx(3sin^ „ sin^ x „ , ^ h2 + C = -sin^ x+sm^ x + C • 2 Chii y rang each va each tren deu cho ket qua dung, no chi sai khdc Tac6:l^=r j d i d u ^ = - J ^ d u = - j d u = - u + C- mot hang so xdc dinh ! b) Thay u = y/l + cos' x ta dugc : I2 = - \ / + cos^ x + C * Cach khac : , /2 = r / Ta CO : ^"^^^ dx = flpi^dx ^ V2sin^x + 3cos^x ^ V + cos^ x (sau dua sinx vdo vi phan) , sm2x , dx = COS" X - r i ^ V2sin^x + 3cos^x Dat u = + =f ^l2 + , ^ =rf(2 + cos''x)- (chu y r / -2cosx r cos^x thuc dao ham - ^ = - (2 + cos^x)2 +C = - V ( + cos^x) + C 2Vx Thay u = + cos^x ta dugc: T = -2V2 + cos^ x - C• x) (2±£^ij0i_ ^ ^ (V^)'=4=)- 2x ^ V + cos^x -1 = - |(2 + cos^x) J ( + cos^x) = _ cong 1/ ==rd(cosx) = — I — = = = = = r d ( c o s ^ V2 + cos^x =^ du = d(2 + cos^ x ) \ f - r = d u = —2Vu+C = Dat u = tan x + va cotx Quan hf giira cos^x— va tanx , s i n ^ X Ta CO ( t a n x ) = — ^ - j - va ( c o t x ) = ] s i n jc cos X vd tana;, sin^ X ^ cos^x r Ta CO : T = - / u d u = i - ^ nen quan he can xet giua —^ cos^x Thay w = t a n x + t a d u g c : b) tan x + b ) ^ sin^ x dx = vi phdn thdnh (atanx + b), dua vdo giica — ^ sin sin^ vdo vi phdn thdnh -(acotx X sin^ x sin'' x c o t x ; -^dx = -(i(cota;) sin X X M = COtX tiiyy tren R) L&i gidi cua bdi todn tan X + + cos 2a; cotx dx \ 3tana; + (3tana; + 4) ODS t a n x + 4) Taco: vd t a n X: a; cos a; dx = - d(3 t a n o a; + 4) o ^ 01'^"^ + 4)irf(3tanx cos X + 4) Dov^ LM gidi cua bai todn l + cos2x (3 t a n X + 4) r 3tanx + dx -J 2cos^x du = d(cot x ) cos X r cot cot" X x , / — ^ a ; = = - J c o t ' a-(i(cot a;) cos X Cdc ban de y quan he giira , dx = sm X smx 3tana; + cos 2x r cot X I dx- a) Phan tick bdi todn : Ta bi6n doi / 12 to CO c o t ' a; — ^ da; = - c o t ' xd(cot a;) • -Do vdy ta c6 thi chon dn phu Id sin' X + b), Gidi mn,ac6 tanx+ 4)^^^ ~ cot^ X — \ • C d c b a n d i y q u a n h e V i d u T i n h : 4- T = cotx sinx cos^x a)/ + c = - ^ + c 12 tick bai todn : Ta biin doi — — d{a c o t x + b ) a (ta hieu cong thuc tren mot each dan gidn nhu sau : dua vai a ^Ovdb tan x + ) ' vd c o t X la : dx = — d{a a d u = d(3 ( t a n x + 4) ^ , - d(3 t a n x + 4) * Nhgn xet: Niu da thgo viec sir dung phuang phdp nay, cdc ban CO thi trinh bay lai gidi nhanh han nhu sau : ,s ^ f ^1 = J = r 33 tt aa n n x + t a n X + 41 ^ X -dx = I : T + cos 2x - ^ cos X r ( t a n x + 4) J 5! 3^^^^^ ^ (3tanx+4f 12 ( ^2 — dx = I cot X - — - — d x — — I cot ss im n sin X X V xd(cot x) ^ = 5—+ \ giai cua bdi toan c - Vay la, chung ta da nghien cihi xong m6i quan he ca ban giup chung ta dinh huang nhanh each giai cho mot bai toan nguyen ham, ciing nhu tich phan sau Trong truoiig hop bai toan khong c6 xuat hien mot moi quan he tren, chung ta lam theo huang giai khac, c6 tinh chat tong quat hon nhu sau: dat an phu u = u(x) de tit nguyen ham theo bien x chung ta bieu dien duac nguyen ham theo bien u! (tiic Id ta can biiu dien bien "x" theo bien u , "dx" theo u vd du) Mb'i cac ban theo doi mot so vi du minh hoa Vi du Tinh : a) thi tit nguyen ham theo biin x chung ta bieu dien duac nguyen ham theo biin u rdi! Vi tit u = x + \ c6 x = u - \ dx = d(u - 1) = du (tiic Id x duac •• biiu diin theo u vd dx duac bieu dien theo du) j x(x + f ' M x ; Dat u = x + =>du = d x v a x = u - l Ta dugc : = J(u - l ) ' u M u = J(u' u 2n' 10 Thay u = x + ta c6 : 1^ = (x + ir - 2u + l)uMu - / ( u ' - 2u'* + u')du u 2(x + 10 l)° ^ (x + 1)^ + C Vi du Tinh : \ ^'('^ + l)'dx • +1 Giai -dx {x-2) a) Phan tich bdi toan : Niu khai triin (x +12)^'"^ rdi nhdn x vac di tinh thi khong kha thi rdi ! O day chung ta cUng khong nhin thdy su xuat hien cua mot moi quan he de dinh huang phep dqt an phu, nhung theo huang giai tong quat, chung ta chon an phu la u = x + 12 thi tie nguyen ham theo biin x chung ta bieu dien duac nguyen ham theo bien u rdi! Vi tit u = x + 12 ta CO X = u — 12 va dx — d{u -12) — du (tuc la x duac biiu diin theo u vd Giai a) Phan tich bdi toan : Trong bdi cUng vay, su dung moi quan he giita x vd x^ khong dem Igi lai giai thoa dang! Neu chon an phu la u = x - thi tit nguyen ham theo biin x chung ta bieu dien duac nguyen ham theo bien u! Vi tit u = X - ta CO X = u + vd dx = d(u + 2) = du (tuc la x duac bieu dien theo u vd dx duac bieu dien theo du) dx duac bieu dien theo du) Lcfi giai cua bdi toan L&i giai cHa bdi toan Dat u = X Dat u = x - 2=>du = d x v a x = u + Ta dugc : + 12 =^ du = dx va x = u - Ta dugc : I = r(u - 12)u^"^Mu = r(u^"i^ - 12u^"^^)du = ' ' J ^ ' 2014 r (u + 2)' + , r u ' + 4u + , + C -9 2013 Thay u ^ X ^ 12 ta CO : I ^ + '^f- _ 12(x + r ^ ^ ' 2014 2013 b) Plian tich bdi toan : Doi vai nguyen ham nay, vice su dimg moi quan he giita xvdx khong dem Igi lai giai thoa dang! Nhung neu chon an phu la u = x -10 + u - ' ^ + u ^ ^ ) d u = r, 11 ^ 9u^ lOn'' ' -1 Thay u ~ x - ta c6 : iW ,M , r, I , , ,, -10 — + -10 u 4u + + C -11 -4 10(x-2r -1 I = ' 9{x-2f -5 + Cll(x-2)" Vi du 10 Tinh : a) I, = f , chung ta * Chuy : Neu dp dung cong thuc (u + 2)^ + l J + 4u + 12 U -1 -4 + -5 :r du T/ -5 10(x-2y" ,sina; + cosx., ( ) cos X cos X X - x^ nen chung (2 + x^)^ Ta CO : r cosx , Ii = / — (sinx + COSx)^ ^ = -dx - {2 + x y x'dx X = -3 ( + x '— ) ' d(2 + x ' ) , vgy ta ^ LM gidi cm bai todn ' ( + x^)^^^^ ^ (2 + x')^ f r cosx = I— ^ ,sinx + cosx.3 , dx ) COS X ( J (tan X Dat u = tan x + xMx= r •-d(2 + x^) - Trong bai todn nay, phdi thong qua mot sSphep bien doi, chung ta mai dp u u = i ( l n | u | + ^ ) + C u Thay w = + xMac6 : I = - ( I n + x' ' = f u-Mu - — + C - ^ + C• J -9 -2 2u' Thay u = tan x + ta c6: i = — 1- C • ' 2(tanx + l f dung duac quan h$ giua u r * Nhan xet: du = d(2 + x^) va x ' = u - -—i—dx= - d ( t a n x + l) + 1)^ cos^ x > ^ , — va — ) va x' + — = x + - LM gidi cua bai todn Dat u = Vx + l =^ \ — nen ta chon anphu la u = x + —• • + X X 1- u +1 ^ _ g r u V - l ) ( u + l ) ( u ^ - u + l)^ dx - = 6j Ta CO X + - =^ du = (1 - -^)dx va x"* + — = x + X x' x' x -2 = u'-2 du _ / ( u - > / ) ( u + V2) u'-2 = ^(lnu-V2 - Inn+ 42 = r 2V2*^lu-V2 + C = -4=ln 2V2 u+ u-V2 U + , u« du u« ^ ^ + C Thay u = Vx + ta dugc : I : = if^i)' V2 l)(u^ - u + l)du = J ( u « - u^)(u^ - u + l)du = j ( u « - u ^ + u « - u ^ + u^-u^)du : du ^u ^gjuV-l)(u^+l) LM gidi cua bai todn Dat u = = X + va u M u = d x {4^if {4^iy {4^if Thay u = x + — ta dugc : +C 1+ l2= — L i n + C = — p r In 2V2 2V2 x + i + ^/J x2-xV2+l +C b) PAa/i ftcA bai todn : Cdc ban di y rdng X V i du 14 Tinh : x+1 xdx (X Vx + + V x + T (7/a/ -1)^ dx x-1 -dx (x - ) ^ Phil Id u = •* , / i T i x-1 ' / ^ + V^l + - - - V x ' + x + - + C \ 2 Vx + + Vx x-1 (x +1)- ta dugc : L = - 2Vx - In V x + T + Vx (Chuy Biiu ndy c6 nghia Id : biiu thuc duai ddu nguyen ham chua u(x) = x - x+1 x-1 vd dgo hdm bdc nhdt cua no u '(x) = Ta chon an phu la u — x+r (X +1)^ L&i gidi cua bai todn ^ r u+1 du (x + l f 1)^ dx dx- ru^ + u^ du f u- + u Suy ra: I = / ' Vx + - Vx] = 2x-2Vx^ + x + Ti^p theo, chung ta su dung phucmg phap d6i bien so de tinh nguyen ham c6 dac dilm : biiu thuc duai ddu nguyen ham chua bieu thuc u(x) vd dao ham bdc nhdt cua no u '(x) Chung ta thuong gap truong hop dac biet: tic thuc Id dao ham cua mdu thuc Khi ta chon an phu la u = u(x) va tat nhien an phu phai dam bao nguyen tSc : nguyen ham moi theo bien u phai de tinh hon nguyen ham ban dau theo bien x Vi du 16 Tinh : x(x-l)Mx b) = / X + -dx ' (x + 1) x(l + xe'') Gidi a) Phan tick bai todn : Cdc ban de y x(x - 1)^ x-1 -dx vd x - dx = X (x + 1)^ x + (X + If X + (X + 1)^ -u + du = (X + u".— = - - / u-1 J -u + l 2^u-l u' + 2u' + 2u^ + 2u + + u - du 21 r (u + + 2u^ + 2u + 2) + _ _ ^ + A + A + 2X + 2u + 21n u - + C 2 _ _ u u 2u + u^ + 2u + In u - + c + —2 + x-1 Thay u x + ta dugc : x-1 fx-1^ [ x - l ] 1 x-l' 1^ = x + 1; x + x + 1; x + + + + C +2 x + l j + 21n x+1 * Cdch khdc ; £)[...]... du 14 Tinh : x +1 xdx (X Vx + 1 + V x + T (7/a/ -1) ^ dx x -1 -dx (x - 1 ) ^ Phil Id u = •* , / i T i x -1 ' 1 > ^ 2 , 1 — va — ) va x' + — = x + - LM gidi cua bai todn Dat u = Vx + l =^ \ — 2 nen ta chon anphu la u = x + —• • + 1 X X 1- u +1 1 ^ 2 1 _ g r u V - l ) ( u + l ) ( u ^ - u + l)^ dx - = 6j Ta CO X 1 + - =^ du = (1 - -^)dx va x"* + — = x + X x' x'... Nhung neu chon an phu la u = x -10 + 4 u - ' ^ + 5 u ^ ^ ) d u = r, 11 ^ 9u^ lOn'' ' -1 Thay u ~ x - 2 ta c6 : iW ,M , r, I 4 , 4 , 5 ,, -10 — + 9 -10 u 4u + + C -11 -4 10 (x-2r -1 I = ' 9{x-2f -5 + Cll(x-2)" Vi du 10 Tinh : a) I, = f , chung ta * Chuy : Neu dp dung cong thuc (u + 2)^ + l 1 J + 4u + 5 12 U -1 -4 + -5 :r du T/ 1 4 -5 10 (x-2y" ,sina; + cosx., 3 ( ) cos X cos X X - x^ nen chung (2 + x^)^... = d u^ -1 2u = 2 \ u^ -1 2u u^ -1 u^ -1 2u 2u r rgc : L = Ta duac -I Thay n — yjx 2n' 2n' du=/ du (l + u)(u" - u' + 2u'(l + u ) 4/ 2 1 u — In 1 1 u + —7 u 2u' 2 u U -1) du u'* du + c + x +1 +C V ^ - - l n f > / ^ + V^l + - - - V x ' + x + - + C 2 \ 2 2 4 Vx + 1 + Vx x -1 (x +1) - ta dugc : L = - 2Vx - In V x + T + Vx (Chuy Biiu ndy c6 nghia Id : biiu thuc duai ddu nguyen ham chua u(x) = x - 1 x +1 x -1 vd dgo ... dugc : + 12 =^ du = dx va x = u - Ta dugc : I = r(u - 12 ) u^"^Mu = r(u^"i^ - 12 u^"^^)du = ' ' J ^ ' 2 014 r (u + 2)' + , r u ' + 4u + , + C -9 2 013 Thay u ^ X ^ 12 ta CO : I ^ + '^f- _ 12 ( x + r... u - ' ^ + u ^ ^ ) d u = r, 11 ^ 9u^ lOn'' ' -1 Thay u ~ x - ta c6 : iW ,M , r, I , , ,, -10 — + -10 u 4u + + C -11 -4 10 (x-2r -1 I = ' 9{x-2f -5 + Cll(x-2)" Vi du 10 Tinh : a) I, = f , chung... tong quat, chung ta chon an phu la u = x + 12 thi tie nguyen ham theo biin x chung ta bieu dien duac nguyen ham theo bien u rdi! Vi tit u = x + 12 ta CO X = u — 12 va dx — d{u -12 ) — du (tuc la