Vay dau la bi quyet de giai nhanh dugc mot bai toan nguyen ham, mot bai toan tich phan noi rieng?. Cuon sach nay viet ra nhSm dem lai cho ban doe nhimg each hieu, nhijng huang di, thu th
Trang 2TRAN TUAN ANH
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
Trang 3GIAI NHANH BAI TOAN
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Xufi't ban nam 2013
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,
Thhy Nguyen Tat Thu, Giao vien Truang chuyen Luang Th6 Vinh
-Bien Hoa - Dong Nai
Tran Tuan Anh
Trang 4G I A I NHANH B A I TOAN T I C H PHAN T R O N G
Trang 5Doi can: x = l = > / = 0; x = 2=i>/ = ln2
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 :
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 c a n : 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
Trang 6V2 2V2-1
T a c o : 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 ( D H k h 6 i D - 2 0 1 3 )
ta CO the gidi nhanh nhu sau :
LM gidi that nhanh gon !
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 trong cuon sach nay !
Trang 7C h L P c n g 1 N G U Y E N H A M
B a i 1 N G U Y E N H A M
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
(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 ( i n x ) =— 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 ' ' l n a hay
= a", tiic la ham so c6 dao ham bac nh^t bang a" Vay ta d l dang
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" !
Trang 8(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) = - s i n x hay ( - c o s x ) = s i n x ,
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
A COS X + sinx.A cos^x + sin^o;
cos^ X cos X cos X
- Vay ham so c6 dao ham bac
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:
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 :
Trang 9a) 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 , I f 1
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 :
V i 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
Trang 10Tiiy theo kha nang cua nguai lam todn met ta c6 the lucre bo di nhirng buac
gidi khong can thiet
Ta bien doi ham so dual ddu nguyen ham ve dang ham cd chira cdc ham
trong bdng nguyen ham co bdn de tinh
Vi du 5 Cho ham s6 f{x) = xe' va F{x) = {ax + b)e'' V a i 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 , V x e R <=><^ , 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 x y
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''
Trang 115 Cho ham s6 / ( x ) = ( x ' + x)e" va F(JC) = (ax' +bx + c)e' Vai gia tri nao ciia
a, b va c thi F{x) la mot nguyen ham cua / { x ) ?
B a i 2 B A N G N G U Y E N H A M M Q R O N G
Sail day chung ta se ma rong cac cong thuc nguyen ham ca ban de dugc
Bang nguyen ham ma rong Bang nguyen ham ma rong la cong cu giup chung
ta tinh nhanh nguyen ham va tich phdn Truac tien ta xet dinh li sau :
1 Dinh li
Nh J f(u)du =F{u) + C vau = u(x) la ham s6 c6 dgo ham lien tuc thi:
J f{u(x)).u\x)dx =F{u{x)) + C
2 Cong thirc nguyen ham mo" rOng
Ap dung dinh li tren trong truong hop u = ax + b{a^0),t^c6 :
jfiax + b)dx= jf{ax + b).{ax + by.-dx
a f(ax + b).{ax + bydx =-F{ax + b) + C
Tom 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 :
Cong t/iii'c nguyen ham cff ban Cong thuc nguyen ham m& rpng
Tinh nguyen ham : I — J (2x + l^dx Neu khong ap dung cong thuc nguyen ham ma rong thi ta khai trien bieu thuc {2x + 1)^, sau do mai ap dung
cong thuc nguyen ham ca ban de tinh :
Trang 12I = J{2x + lydx = J(2x + l)(2x + Ifdx
= J {2x + l)i8x''+12x^+6x + l)dx
= J (16a;' + 24a;' + 12a;' +2x + 8x'' + 12a;' + 6x + l)dx
= J (16x' + 32a;' + 24a;' + 8a; + l)dx
16x' 32x* 24x' 8x'
+ Sa;" + 8a;' + 4a;^ + a; + C
16x-'
5
Bay gia chung ta ap dung cong thuc nguyen ham ma rong (cong
thuc r{ax + bydx = - ^"•'^ ^ + C(a ^ - 1 ) ) d^ tinh/, ta c6 :
^ o a + 1
«^ ^ ^ 2 4 + 1 10
(C/zw >• rang, each nay va each tren deu cho kit qua dung, no chi sai khdc
nhau mot hang so xde dinh!)
Neu bai toan tren ta thay s6 mu 4 bang so mu 2013 chang han thi lai giai
nhu each dSu tien se phuc tap nhu the nao? Con neu chung ta ap dung cong
thuc nguyen ham ma rong ta c6 ngay lai giai ngan gon cho bai toan do la
/ = / (2a; + If^^dx = ^ '— + C Ke ca chung ta dung phuang phap
J 4028
d6i biln s6 (se hoc a bai sau) thi cung c6 lofi giai khong gon bang each nay !
R6 rang cong thuc nguyen ham ma rong to ra uu diem han cong thuc nguyen
ham ca ban ! Cac cong thuc nguyen ham ma rong, neu chung ta cho he so a =
1; i = 0 thi ta thu dugc cong thuc nguyen ham ca ban
Trang 13•dx = —.- - 1
a (n-l){ax + b) — + C{n^l;a^O) se cho
ta lai gidi nhanh han nua (vi gidm duac mot buac biin doi) !
- Neu khong dp dung cong thuc nguyen ham ma rong, chung ta gidi bdi
todn bang phuang phdp doi bien so md chung ta se xet trong bdi hoc sau O
day chung ta dan cu cdu a) duac gidi bdng phuang phdp doi bien so de so
sdnh hai cdch gidi:
/ = 2a; + 3 dx 12
1 Dat M = 2a: + 3 => (iti = 2dx rfx = - du
Rd rang cdch nay to ra khd phuc tap so vai mot bdi todn dan gidn nhu vdy !
Ap dung cong thuc nguyen ham ma rong ta c6 lai gidi gon vd nhanh han:
\13
12
2a; + 3 dx = 2x + 3
26 + C
Do vdy, viec nha vd van dung tot cong thuc nguyen ham ma rong Id can
thiet de chung ta tinh nhanh duac nguyen hdm vd tich phdn sau nay
d) 7 = fe-''dx = —e-'+C = -e-^+C
- Chung ta c6 the trinh bdy nhanh nhu sau :
Trang 14; d) ^ 4 = / 2cos(3 )rfa;
Bai 3 PHl/QNG PHAP D O I BIEN SO
Bai nay chiing ta se xet hai truong hop Ichi tinh nguyen ham /{x)dx bang
phuong phap doi bien so :
- Truong hop 1 : Dat u Id mot ham so cua x
— Truong hop 2 : Dat x la mot ham so cua u
A Phep dat u la mot ham so cua x : u = u(x)
Gia su can tinh / = /{x)dx, ta thuc hien nhu sau : Buoc 1 : Chon an phu thich hop u = u(x)
Buoc 2 : Xdc dinh viphdn du = du(x) hoac du^ - du^(x)
Buoc 3 : Bieuthi f(x)dx theo u va du Gidsurang f (x)dx = g{u)du Buoc 4 : Tinh I = g(u)du Sau do thay u = w(x) de dime ket qua can tim
Chu V : chon an phu u = w(x) sao cho viec tinh / = g(u)du phai de hon la
tinh / = jf(x)dx !
Khi nhin vao mot bai giai cho bai toan tinh nguyen ham 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? 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 nguyen ham, tich phan cua cac ban
Truoc tien cac ban c^n luu y hai ket qua ma chiing ta thuong dung sau day :
(1) df{x) = f\x)dx
Trang 15(2) Niu J f(u)du =F{u) + C va u = u(x) la ham sS c6 dqo ham lien tuc
thl: J f{u{x)).u\x)du = J i\u{x))du{x) =F{u{x)) + C
Vi du
a) J cos(2x^ + 3a; + l)d{2x^ + 3a; + 1) = sin(2a;^ + 3a,' + 1) + C
(ta hieu trong suy nghi " 2x^ + 3x + 1 " lau)
b) f ^- d(x' + 1) = l n a;' + 1 + C = ln(a-' + 1) + C v/ + 1 > 0 (ta
^ (a;' + 1 )
hieu trong suy nghi " -\-\" la u )
Sau day chiing ta tim hi6u cac moi quan he quan trpng giup chung ta tim
nhanh phep dat in phu va dinh huong nhanh each giai cho bai toan nguyen
ham, tich phan bang phuang phap doi bien so
l.Quan hegiua x" va x"*\n^-\)
Ta CO : dx'"' = (« + \)x"dx o x"dx = —dx"*' '
n + \ + \)
d{ax"^'+b), trong
do a^O con b tuy y tren R Vay ta c6 quan he giira x" va x"^\n^-1) nhu
(ta hieu cong thicc tren mot each don gidn sau : x"dx = -
1
a{n + \) d{ax"''+b) nhu sau : dua x" vdo trong vi phan thl thdnh {ax"*^ -^b), voi a ^ Q vd h tuy
ytren
V i d u l T i n h :
a) Jx^{2x^ +lfdx\ = Jx4^+ Idx
Giai a) Phan tich hai todn: Theo I6i giai thong thuang, cac ban se khai trien
bieu thuc (2x + 1) , sau do nhdn vai X de dua ve nguyen ham de tinh han
The nhung viec khai trien bieu thicc (2x^ + if la khong dan gidn? Do vdy,
each nay da to ra khong hieu qua ! Niu giai bai todn nay bang phuang phap
doi biin so, ta chon an phu la u-lx^ + \ Tgi sao Igi chon duac an phu nhu
vay? Bay gia cac ban de y quan he giica x^ va x^ nhu sau :
x^dx = -d(2x^ +1), nen ta c6 x'{2x^ + ifdx = -(2a;' + lfd{2x^ + 1)
' 60
b) Phan tich bai todn: Cac ban de y quan he giica x vd x :
xdx = - d(x^ +1) nen ta c6 x^x^ +\dx = - Vx^ +1 d(x^ +1) Do vdy, ta c6 thi
2 2
chon an phu Id w = +1 hodc u = Vx" + 1 Trong truang hap nay ta nen chon
u = de bieu thuc dual ddu nguyen ham khong con can thuc
L&i giai cHa bai todn