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Giải nhanh bài toán nguyên hàm tích phân dành cho học sinh lớp 11 -12

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  • Giải nhanh bài toán nguyên hàm  tích phân dành cho học sinh lớp 11 -12

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515.076 GI-103N TRAN TUAN ANH Gliil NHANH BAI TOAN HGIIYJN HAH &TiCH PHAN DANH CHO HQC SIHH L0P11-12 TRAN TUAN ANH m r m m m • 1 » - ^ NGUYEN HAM ^TICH 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 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 CONG TY TNHH M^T THANH VIEN SACH VIET . 391/15A Hajnh Tin Phat, P.T§n Ttw^n Dong, QuSn 7, TP.HCM, BT: {06} Jf.720.837 • Fax P) 38.726,052 • MST: 03114307135 Email: WifiaclRfietcoxom- Website: «»w.sachvie!co.«ni Xufi't ban nam 2013 La>i noi dau Viec giai mot bai toan noi chung la mot qua trinh tu duy 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 nay den each khac mai giai xong, trong khi 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 nay viet ra 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 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 nay. 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 thanh 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 - Truang DH Kinh Te Luat. i ' ' - Thhy Nguyen Tat Thu, Giao vien Truang chuyen Luang Th6 Vinh - Bien Hoa - Dong Nai. Tran Tuan Anh GIAI NHANH BAI TOAN TICH PHAN TRONG DE THI TUYEN SINH DAI HOC NAM 2013 Cau 1; Tinh tich phan / = |—~ 1" • (E>H kh6i A, Ai - 2013) Cdch 1: /= ——.lnxdx= \nxdx + Cdch gidi thong thir&ng 2 w -I 1 In xdx. Ta xet: + A = In xdx. Dat u = lnx=> du = —dx ; dv = dx=> v = x. X In xdx = x\nx 2 ' 1 -dx = 2\n2-\. X + h = -1 In xdx Dat u = \nx^ du = — dx; dv = X -1 In xdx = \X J f 1 \ , 1 rfX => V = — . X 1, = — Inx X 2 ^ 1 f 1 dx X = —Inx X 2 I + - 1 X = -ln2—. 1 2 2 Vay / = /,+/2=21n2-l + -ln2-i = -(51n2-3). V-1 . In xdx - 1- 1 Inxc/x Dat /• = In X => X = e' va dx = e' dt. Doi can: x = l=>/ = 0; x = 2=i>/ = ln2. In 2/ , \n 2 In 2 1= ^\\—L-\e'dt= \[e'-e")dt= \td(e'+6-) 0 V = t(e'+e-) In 2 0 In 2 • '(e'+e-')dt = ln2. V 2, 2-i 2 .l(51n2-3). Vay / = |(51n2-3). Cac/i ^w/ nhanh Cdch 3: cdc ban di y quan he giita — va \ : -^dx = d X X X ; quan he giiia X vd 1 la : \dx = dx. Do do, ta CO : ^x'-\^ dx = dx = d x + - Vdy ta CO the giai nhanh bdi todn tren nhu sau : 2 2 1 = x'-\ . In xdx = 1- 1 ^ = ln X. x + - 2 V O - x + - 1 rv In xdx = In xd * 1 X x + - = Inx. \^2 x + - 2 1 + 1 ^ dx = In X. r 0 2 ( 1' x + — — X — 1 = |(51n2-3). I,^/ giai that nhanh ggn so v&i hai cdch tren ! Cau 2; Tinh tich phan / = ^x^l-x'dx (DH kh6i B - 2013 Cdch giai thong thu&ng Cdch 1: Do dau hieu " nen ta chon an phu x = V2 sin?. Dat X = y/l sin t => dx^-Jl cos tdt, te n n Doi can: x = 0 / = 0;x = 1 => / = —. 4 Taco: /= JV2 sin / V2 - 2 sin" /.72 cos / Jr = 2 V2 Jsin cos Vl - sin^ / J/ = 27^ sin ^ cos/. cos/L// = 2V2 sin/cos tdt. Xet tich phan J = 272 |sin / cos^ tdt. Dat u = cos/ ^du = -s'mtdt. K 72 Doi can : / = 0 =o w = 1;/= — => w = —. 4 2 2 1 3 Taco: J = -l4l \u^du = 2^ \u^du = 2^.— 2 7^ = 272-1 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. D6i can:' x = 0 => / = 72;x = 1 =i> / = 1. V2 2V2-1 Taco: I = -\t^dt= \t^^t= — Cdch gidi nhanh Cdch 3: Cdc ban de y quan he giua x vd x^ la: xdx = ~^d{^x~^ = -^d{2-x^y Nen viec ta chon an phu t (0 cdch 2) la hodn todn tu nhien ! khong mang tinh dp dat cua kinh nghiem trong suy nghi Id : "thay cd can thiic thi dat can thitc la an phu". Chung ta c6 the gidi nhanh nhie sau: 3 x42^dx = ^\2-xjd(2-x') = ^^^-p!- 1 _ 2V2-I 0" 3 L&i gidi that nhanh gon ! Cau 3: Tinh tich phan / = .(x + 1)^ , -<lx (DHkh6iD-2013) x +1 Cdch gidi thong thirong Cdch 1: Ta c6 : / = (x + 1)' + U V. V 2x x'+\ x'+\ -dx = dx + 0 0 -dx = \ r 2x -dx Xet tich phan J = 2x x'+\ •dx. Dat / = +1 =i> J/ = 2xdx. D6ican: x = 0 / = l;x = 1 / = 2. Tadugc J= f—= ln = ln2. Vay / = l + ln2. Cdch 2: Ta c6 : /= ;—'—dx= ; dx^ \dx+ \— dx^\ i X +\ 1 J J r- -4-1 J Xet tich phan J = 2x x^+1 x^+1 •dx. 0 0 2x_ x'+\ V 2x x'+l -dx Dat x = tan/=>(ix = —^—dt = {\-^ian-t]dt, ti cos"/ ^ ' Doican: x = 0 / = 0;x = 1 =^/ = —. 4 2 '2 Tadugc /= f^^(l + tan^/)^/ = 2 f^c// tan / + 1 V 1 = -2 : of (cos/) = -2 In J cos/ 1 sin/ < cos/ cos/ 4 =-2 In 0 4i' Vay / = l-2hi-^ = l + ln2. V2 Cdch gidi nhanh Cdch 3: Cdc ban de y quan he giua x vd x^ Id: 2xdx = ci(x^ j = (x^ +1). Nen viec ta chon dn phu t = x' -\-\(o cdch 1) Id hodn todn tu nhien ! Chung ta CO the gidi nhanh nhu sau : 1 = Ax + \)\+2x4-1 -flX = I 1 x^+1 , dx = \dx + f ^"^ dx 0 0 = X + x^+1 i/(x'+l) =x 0 + ln x^+l = l + ln2. LM gidi that nhanh gon ! D6 CO each nhin "tudng minh" vh each giai nhanh Nguyen ham va Tich phan, mai ban doc tim hieu nhirng kien giai trong cuon sach nay ! ChLPcng 1. NGUYEN HAM Bai 1. NGUYEN HAM 1. Dinh nghIa 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 F'(x) = f(x) vai mgi x thugc K. Mgi ham s6 f(x) lien tuc tren K d^u c6 nguyen ham tren K. Sau nay, yeu chu tim nguyen ham cua mot ham s6 dugc hieu la tim nguyen 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 = F{x) + C Vi du 1 a) J2xdx = x^+C vi (x'+C)' = 2x. b) cosxdx = smx + C vi (sinx + C)' = cosx. * Luu y: di hiiu nhanh nhung noi dung kien thuc trong cuon sdch nay, ban doc nen ren luyen thdnh thgo viec tinh dgo ham ! 2. Tinh chat thii- nhat f'{x)dx=fix) + C Tinh chat thu nhSt dugc suy true tilp tir dinh nghia nguyen ham. Trong thuc hanh, tinh chk nay giup ta tim ra nguyen ham cua mot ham so don gian, cung nhu viec xac dinh lai nguyen ham tim ra 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 cm no phdi chinh la f(x)'\i each hieu do, chung ta c6 the thanh lap Bang cong thuc nguyen ham co ban nhu sau : (1) Cong thirc 1 : Qdx =? Ta suy nghi : ham so nao c6 dao ham bac nhat bang 0? Hien nhien do la hang so ! Vay ta c6 cong thuc thii nhSt: Qdx = C (2) Cong thii-c 2 : \dx=l Ta suy nghi : ham so nao c6 dao ham bac nhat bang 1? De dang nhan thay do la X vi x' — 1. Vay ta c6 cong thuc thu haii \dx = = x + C (3) Cong thii-c 3 : x"dx =? Ta suy nghi: ham s6 nao c6 dao ham bac nhat bang jc"? Chung ta lien tuong ngay toi cong thuc dao ham {x")' = nx"'^ hay = x" . Ta thay n-\^a hay « = a +1, thu dugc cong thuc « + l = X hay = x" . Vay la ham so c6 dao ham bac nhat bang x". Suy ra a + \ cong thuc thu ba : a+\ x"dx=-— + C ar + 1 (a^-1). (4) Cong thuc 4 : f—c/x =? Ta suy nghi: ham so nao c6 dao ham bac nhat bang — Ta lien tuong toi cong thuc (inx) =— thi thu duac cong thuc X 9 X \-dx^\nx+C J V . Chiing ta lay dau gia tri tuyet doi vi dieu kien ciia ham Logarit! (5) Cong thij-c 5 : 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''lna hay = a", tiic la ham so c6 dao ham bac nh^t bang a". Vay ta dl dang In a vlna; thu dugc cong thiic a a'dx = — + C \na (a>0,fl^l). (6) Cong thuc 6 : e''dx=? Ta suy nghi: ham so nao c6 dao ham bac nhSt J bang e"') De dang ta nhan thay do la ham e' vi (e'') =€' , suy ra cong thiic thu sau : e^dx = 6" + C . Cong thuc thii sau la truofng hgp rieng ciia cong thiic thii nam khi thay "a" bang "e" ! (7) Cong thu-c 7 : jcosxdx =? Ta suy nghi : ham so nao c6 dao ham b$c nhk bang cosx? Tir cong thuc quen thuoc (sinx) =cosx, ta c6 ngay cong thuc thu bay la : cosxi/x =sinx + C (8) Cong thii'c 8 : sin xdx =? Ta suy nghT: ham so nao c6 dao ham bac nhat bang sinx? Tu cong thuc quen thupc (cosx) =-sinx hay (-cosx) =sinx, ta CO ham so ma dao ham bac nhat cua no bang "sinx" la "- cosx", suy ra cong thuc thu tarn la : | sin xdx = - cos x + C (9) Cong thuc 9 : 1 1 cos X -dx=1 Ta suy nghT : ham so nao c6 dao ham bac nhat bang —r— ? Truang hop nay khong de tim nguyen ham hon cac truoTig cos X hop tren ! chung ta dir doan ham so can tim c6 dang (chu y do mau thuc cos a; "cos^x"). Ta c6: cos a; A. cosx + sinx. A , ro rang neu chon A = sinx thi cos X A. COS X + sinx.A cos^x + sin^o; cos^ X cos X cos X Vay ham so c6 dao ham bac ' ^ 1 ' Sill X nhat bang —-— la ham so hay tan x. Suy ra cong thuc thu chin : cos X cos a; r —Y~ — x + C ^ cos X (10) Congthiic 10 <-' sin' •dx = ? Ta suy nghi : ham so nao c6 dao ham sm X bac nhat bang ? Tuang tu cong thuc 9 ! Minh du doan ham so can tim CO dang (chu y do mau thuc "sin^ x "). Ta c6: sinx smx la sm^ X sin^ X — sin^ X — cos^ X sin^ X ^ . Vay ham so c6 dao ham bac nhat bang ^ sin^ X sin^ X ham so - cos X sinx hay (- cotx). Suy ra cong thuc thu muai: sin^ X dx = - cot X + C Vay ta c6 Bang nguyen ham ca ban sau : Jodx = C f adx = + C{cy > 0; Q ^ 1) In a J dx = X + C y cos xdx — sin x + C / x"dx = - + C(a ^ -1) J sin xdx = — cos x + C f — dx = \n X +C X f —^-— dx = tanx + C ^ cos X J e'dx = e' +C f —— dx = — cot X + C sin^ X Hieu va thuoc bang nguyen ham ca ban la dieu kien thiet yeu de chung ta tinh dugc nguyen ham cung nhu tich phan sau nay. Chinh vi vay, chung ta can su dung thanh thao cac cong thuc trong bang nguyen ham ca ban. 3. Tinh chat thu- hai J kf{x)dx = kj f{x)dx Trong cong thuc nay, dieu ma chung ta can chu y la he s6 "k" (he so k c6 the "ra", "vao" qua dau nguyen ham!), tat nhien k phai la hang so, con bien so khong dua ra ngoai dSu nguyen ham dugc. Vi du 2. Ap dung tinh chSt thu hai va Bang nguyen ham ca ban, ta c6 : a) J 6xdx = GJ xdx (dp dung tinh chat thu hai) = 6 — + C (dp dung bang nguyen ham ca ban) 2 = 3a;' + C . , . r cos X , If 1 cos xdx = — s inx + C • 3 c) j e^^'dx = J e.e'dx = e J e'dx = e.e' + C. d) \mx\lx = \Q\x'dx = \0.^— + C = 6x' +C. - + 1 3 Noi chung, khi tinh nguyen ham cung nhu tich phan sau nay, chung ta c6 gang bidn d6i ham s6 dual dau nguyen ham hay dual dau tich phan xuat hien nhung ham s6 c6 trong bang nguyen ham ca ban. Do vay, viec nam dugc Bang nguyen ham co ban la di^u kien rSt quan trong de chung ta tinh dugc nguyen ham, tich phan. 4. Tinh chat thir ba J" {J{x) ± g{x))dx = J f{x)dx ± J g{x)dx Chung ta c6 the hieu mot each dan gian cong thiic tren nhu sau: nguyen ham ciia tong (hieu) cua hai ham so, bang tong (hieu) cac nguyen ham cua hai ham so do. Cong thuc CO the ma rgng nhu sau : / U^{x)±f^(x)± ±l{x))dx=^ f^{x)dx± f f^{x)dx± ± J l{x)dx Bay gia chiing ta di xet cac vi du minh hoa : Vi du 3. Ap dung cac tinh chat va Bang nguyen ham ca ban, ta c6 : a) = J {4x + 3cosa;)(ia; = J Axdx + J 3cosa;c?x- (dp dung tinh chat thu ba) = 4 J" xdx +3 J cos xdx (dp dung tinh chdi thu hai) = 4 h 3 sin a; + C (dp dung bang nguyen ham ca ban) • 2 = 2a;^ + 3 sin X + C b) / = r (5e" ^)dx = r Se'dx- f dx (dp dung tinh chdt ' cos' x *^ thif ba) cos' x = 5 r e^'dx - 7 r —-— dx (dp dung tinh chdt thu hai) . "J ^ cos' X = 56^^-7 tan x + C . (dp dung bang nguyen ham ca ban) ^ ' . f 1 . '^-^-Vx+ [x'dx c) I, =: j 3- +^ c/x= J3-Vx+ \-jr^dx = \l>\y y <lx J \x I = 9{ydx+ {x'dx = — + ^ + C = -— + 3x'+C. J J ln3 1 ln3 3 3x' - 2x + 4 dx = X I Sx' 2x 4 + - XXX dx J Sxdx - J 2dx + j : 3Jxdx-2Jdx + AJ-dx ^ — -2x + A\nx + C. X 2 Trong thuc hanh, ta trinh bay nhanh nhu sau : a) I, = Ji4x /(5e c) I, \ J 33;' 3x' -^)dx =5e^ - 7 tan x + C. cos X -dx X 3\3'+x' -2\ i cix = — + 3x'+C. in 3 3x - 2 + - X dx 23; + 4 hi X + C. Tiiy theo kha nang cua nguai lam todn met ta c6 the lucre bo di nhirng buac gidi khong can thiet. Vi du 4. Tinh : ^ X c) = j T.ydx; d) h = X dx -dx. 42 Gidi Ta bien doi ham so dual ddu nguyen ham ve dang ham cd chira cdc ham trong bdng nguyen ham co bdn de tinh. a) Ta CO : -I X + 1)^ dx X -I X dx X 1 + 2 1 2 1 1 + —+ - X ^ dx = X- + 4^2!^ + In X + C. dx b) Ta CO : 7^ = J X + - e^x^ •dx x x e'x^ 3 3 ,3 JC JC dx -I 1 1 x X dx = —- + In X X ~e'' +C. c) Taco ': 73 = JT.2,'dx = J(2.3)'dx = jG'dx = — + C. In 6 re d) Taco : = \—;<ix = 4' 42 V / -dx = ce -dx-^ dx = e v2y V ^ / In 'e^ v2/ + C. Vi du 5. Cho ham s6 f{x) = xe' va F{x) = {ax + b)e''. Vai gia tri nao cua a va b thi la mot nguyen ham ciia f(x) 9 Gidi Tap xac djnh cua F(x) va /(x) la R. Ham s6 F(x) la mpt nguyen ham cua f(x) thi /^'(^) = fix) vai Vx e R . Ta CO : F'(x) = ae" + (ax + b)e''(ax + a + b)e" nen F'(x)=-f(x) vai Vx e R thi (ax + a + b)e' = ' vai Vx e R <=> ax + a + 6 = x, Vx e R a-l = 0 |'a = l <^(o-l)x + a + Z)-0,VxeR <=><^ , Ci>< . Vay vai a = l va a + b-0 b = -l b l thi F(x) la mot nguyen ham cua f(x). Vi du 6, Chung minh rang F(x) - sin xe"" la mot nguyen ham cua ham so /(x) = (sin X + cos xy . Gidi Tap xac djnh cua F(x) va /(x) la IR . Taco: F'(x) = (sinx)'^''+ sinx(e'')' = cos xe"" + sin xe' = (cos x + sin x)e'' = /(x). Vay F(x) = sin xe' la mot nguyen ham cua ham so /(x) = (sin x + cos x)e''. (dpcm) BAITAP 1. Tinh : 25a.'' +122'+ 1991 X dx; X dx; b) 7, = J 15a; + 10V^ + 1983 da;. 2. Tinh : a)= J'^Ssina; — 4cosa;jda;; b) 7^ = J" tan a; a;' - 2a; + - X da;. cos X sm X dx . 'J SI sin 2x -dx', [...]... hay tich phan bang phuong phap dat an phu (hay phuong phap doi bien so), ban doc thuong c6 cau hoi : tai sao lai chon dat an phu nhu vay? Lam sao chon an phu thich hop? ^(41 + 5) Nhirng Icien thuc duai day se giiip cae ban dinh huong dugc phep dat an phu cho minh mot each nhanh chong ma Ichong phai may mo lam giam toe do tinh 2 Tinh: b) = J nguyen ham, tich phan cua cac ban Truoc tien cac ban c^n luu... tick bai todn : Ta nhln thdy ddu hieu g(u)du an phu x = ip{u) sac cho viec tinh / = jg(u)du phai de honti la tinh / = | / { x ) d x ! Cac dSu hieu dan tai viec lua chon i n phu theo kiku tren dugc cho duai bang sau: + ^\u)du J * Cliiiy : chon < 2 2 cot'u (0 < M < TT) V i du 1 Tinh : Birac 1: Chon an phu thich hap x — ip{u) (theo ddu hieu cho trong bang duai) dx = d^f{u) — — a ( TT, Chung ta xet cac v... han vd khong phuc tap nhu cdch dat u = cosx Vidu 11 Tinh: a) I, = b) rdX ' vgy ta chon an phu la u = cosx hoac u = Vcosx Trong truang hap nay chung ta nen chon u = Vcosx de bieu thuc duai ddu nguyen hdm khong con chica can thuc • = J xVl + xdx Gidi a) Phan tick bai todn : Bai todn ndy sit dung 7 quan he de dinh huang phep dat dn phu Id khong khd thi Neu chon an phu Id u = V l — x ihi tit nguyen Ldi giai... + 1 + 1 1+ 1 1+ -1 x ' — hi Vx + 1 + 1 - h i hi Suy ra = in I x I - hi I +1 + 1 Vx'^ + 1 + 1 1= 111 1+ x -1 = * Cach khac : Ta chu y bien doi 4x ' +1 = In 1 •dx = XV x^ + 1 • dx va sir x^V? + 1 dung quan he giua x^ va x ta chpn hn phu la u - y x ^ + 1 Vay lai giai khac cho bai toan la : I - Hai each deu cho ia kei qua dung: 1+ ^ 2 I + 1 =4> udu = xdx Thay u = Vx' + 1 ta dugc: sin^ u + cos^ u = 1) Vay... trinh bay lai gidi nhanh han nhu sau : ,s ^ f ^ = J 1 = 1 r 3 tan x + 4 3 tan tan X + 4 ^ X -dx = I : T + cos 2x - ^ 2 cos X 3 r ( 3 t a n x + 4) 1 J 5! 3 3^^^^^ ^ (3tanx+4f 12 ( ^2 — dx = I cot X - — - — d x — — I cot s im s n sin X X V xd(cot x) ^ 3 = 5—+ \ giai cua bdi toan c - Vay la, chung ta da nghien cihi xong 7 m6i quan he ca ban giup chung ta dinh huang nhanh each giai cho mot bai toan nguyen... 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 do 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 nay 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 - 2 thi tit nguyen... 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 + 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)^ +... ) + C ai s i ln ' ( 2 x - l ) f (x)dx = g{u)du Chu V : chon an phu u = w(x) sao cho viec tinh / = g(u)du phai de hon la 1 Tinh : = theo u va du Gidsurang Buoc 4 : Tinh I = g(u)du Sau do thay u = w(x) de dime ket qua can tim BAITAP a) Buoc 3 : Bieuthi f(x)dx J (4x + 2)' dx; tinh / = jf(x)dx b) ^ = / - ^ ^ - ; ^ (3-2x) ! Khi nhin vao mot bai giai cho bai toan tinh nguyen ham hay tich phan bang phuong... 1 //zeo tu la can bac hai, can bdc ba Ma 6 la boi chung nho nhdt ciia 2 va 3 nen ta chon dn phu Id u = yjx + 1 hay x + 1 = u'^ de lam mdt can thicc 1 = 1—\ (chu y quan he giua x+ - cho x^) Tai day cdc ban lai de y rang X 1 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 ^... lai, ta c6 cong thuc dk mo rong bang nguyen ham ca ban: '/{ax + b)dx = -F(ax + b) + C a Cong thuc nguyen ham ca ban va cong thuc nguyen ham ma rpng dugc cho tuang ung duai bang sau : truang hop viec ap dung bang nguyen ham ma rong cho ta lai giai bai toan nhanh va " sang " han ! Chang han vai bai toan sau : Tinh nguyen ham : I — J (2x + l^dx Neu khong ap dung cong thuc nguyen ham ma rong thi ta khai trien . — e-'^'^'^^di-?, sin x + 2)- 3 Do vdy, ta chon an phu Id u = -3sinx + 2. LM gidi cua bai todn = Jcosxe-'''"''^'dx = J^e-"'"'^+'(i(-3sinx . ax + b a J e'dx = +C f e'"'-'dx = -e'^^'' +C 'J a f adx = — + C{a>Q]a^ 1) In a ra'^^'dx = ±.^ + C{a > 0;a ^ 1). so. l.Quan hegiua x" va x"*^-) Ta CO : dx'"' = (« + )x"dx o x"dx = —dx"*' ' n + + ) d{ax"^'+b), trong do
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