Nội dung

Cty TNHH IVITV DVVH Khang Vi§t ca'p t6'c giai 10 chuyfin dg 10 diem thi m6n Toan - Nguygn Phu Khanh Chuyfin Ta CO- f'(t) = 3t^ + l + - ^ > 0 ^ ' tln2 voi V t > 0 , do do f(t) luon dong bien trong khoang (0;+oo), phirang trinh (3) NGUYfeN H A M , TiCH PHAN VA UNG DyNG f(x) = f(2y) tux x =-2y De hoc gioi mon tich phan lap 12, cAc em can: 1. Thuoc bang dgo hdm ( nguyen hdm la ngwo-c v&i dgo ham ) ; 2. Hieu rd vd nam ki cdc dinh ly tich phan; 3. Lam nhieu bai tap vt nam vu-ng cdc dgng todn. 1 1 Vai X = 2y, phu-o-ng trinh ( 2 ) tra thanh : y^ = - o y = - thoa y > 0 . Vay, he cho c6 nghiem: (x;y) = IV (4] b. Dieu kien: xy > 0 . Dat t = log2 (xy) => xy = 2', khi do phu-o-ng trinh ( l ) tro thanh: 9' - 3 = 2(2')'°''' o 3^' - 3 = 2.3* « 32* - 2.3* - 3 = 0 1. Dinh nghia: Cho ham so f xac djnh tren K . Ham so F du-gc goi la nguyen ham ciia f tren K neu F'(x) = f (x) Vx € K . 2. Cactinhchat: Djnh li 1. Neu F la mot nguyen ham ciia ham f tren K thi moi nguyen ham cua f tren K deu c6 dang F(x) + C, C e l . Do vay F(x) + C goi la ho nguyen hamcuaham f tren K va du-gc ki hieu: f(x)dx = F(x) + C. < » ( 3 * + l ) ( 3 * - 3 ) = 0,suyra 3* = 3 tu-c xy = 2 ( 3 ) . Phu-ong trinh (2) « + y^ + 2x + 2y +1 = 0 o ( x + y f + 2 ( x + y)-2xy + l = 0<::>(x + y f + 2(x + y ) - 3 = 0 do (3), Djnh li 2. Moi ham so lien tuc tren K deu c6 nguyen ham tren K Djnh h' 3. Neu f, g la hai ham lien tuc tren K thi: I. + j i f ( x ) ± g ( x ) ] d x = j f ( x ) d x ± jg(x)dx. + k.f(x)dx = k f(x)dx vaimoisothirc k^tO. phu-o-ng trinh nay tu-o-ngdu-ong (x + y - l ) { x + y + 3) = 0 Djnh h' 4. Neu ff (x)dx = F(x) + C thi <=>x + y = - 3 hoac x + y = l THI: TH2: • x + y = -3 xy = 2 x+y=1 „ Jx=2 <»< hoac • y=.l ly = 2 ' •f{u(x)).u'(x)dx = Jf (u(x)).d(u(x)) = F(u(x))+ C. |x = l 3. Bang nguyen ham cac ham so thiro-ng gap Cac ham so cap thu'o-ng gap Ci + tru-o-ng hop nay v6 nghiem. l «dx = ^ xy = 2 a +1 dx Vay, he da cho c6 nghiem ( l ; 2 ) , ( 2 ; l ) . = ln X + C (a^-1) ^ ' +C e''dx = e''+C •a''dx = — + C Ina + • = - l n ax + b +C ax + b a sin(ax + b)dx = -i.cos(ax + b) + C + |cos(ax + b)dx = -.sin(ax + b) + C • = -tan(ax + b) + C cos (ax + b) a sinxdx = -cosx + C cosxdx = sinx4-C dx cos^x r dx sin^x Nguyen ham mo- rong dx sin (ax + b) dx :tanx + C = -cotx + C Vax + b = —cot(ax + b) + C a =—vax + b + C a cap tOc giai 10 chuyen flg 10 d i l m thi mfln Toan - Nguygn Phu Khanh ' ' Vay, F(x) = ( 4 x ^ + x + l l ) V 2 x - 4 . f 3X + b i(cx-a)(dx-p) Vi du 2. T i m nguyen ham : Tach phan thu-c trong tich phan t r a thanh: p • 3 X 4" b Lay nghiem cua c x - a thay vao - — - ta diro-c p dx - p • Lay nghiem cua dx - p thay vao 3X +b +q cx-a -dx (x + l f I2 = fsin3xcos5xd}< Lo-i gidi 1. ta du-p-c q cx-a X li = 1 -dx = h-j- d(x + l ) I 1 Dang 1: T i m nguyen h a m bang phu-o-ng phap phan tich Vi du 1. 1 3{x+iy 2. 4{x+iy - + C. f I2 =Jsin3xcos5xdx = ^ J ( s i n 8 x - s i n 2 x ) d x = -^ cos8x + cos2x 8 2 +C 1. Go! F(x) la nguyen ham cua ham so f ( x ) = sin2x.tanx thoa man F Dang 2: T i m nguyen ham bang phu-cng phap doi bien so Tinh F l4j 2. Xac dinh a, b, c sao cho F(x) = (ax^+bx + c ) V 2 x - 4 ^ . 20x^-29x + 7 ^ ham so f ( x ) = . — /_ Neu | f ( x ) d x = F(x) + C t h i j f ( u ( x ) ) . u ' ( x ) d x = F ( u ( x ) ) + C". la 1 nguyen ham cua Gla su- ta can t i m ho nguyen ham 1 = |f ( x ) d x , trong do ta c6 the phan tich V trong ( 2 ; + 0 0 ) . f ( x ) d x = g ( u ( x ) ) u ' ( x ) d x t h i t a thu-c hien phep doi bien so t - = u ( x ) \/2x-4 Lai giki =^ dt = u ' ( x ) d x . Khi do: I = j g ( t ) d t = G ( t ) + C = G ( U ( X ) ) + C 1. T a c o : F ( x ) = f s i n 2 x . t a n x d x = f 2 s i n x . c o s x . - ^ ^ d x = 2 fsin^xdx ^ ^ J J cosx •' F(x)= |(l-cos2x)dx = x - ^ ^ Chii y: Sau k h i ta t i m du-ac ho nguyen ham theo t t h i ta phal thay t = u ( x ) . Vi du 1. T i m nguyen ham: +C 1. Ma: Vay: F 7 I 1 . 2 7 C _ N / 3 y(3 •• — o 4 3 r./ \x yj3 n l3 F(X) = X 4. = 2 2 + ^ s m — + C = — =>C = 3 4 2 2 3 2. I2 = | x ^ V x 2 + 9 d x n Lo-i g i i i 3 1. I j = |xVx + l d x . 1 W c h l : D a t t = x + l = > x = t - l va dx = dt 7t 1 . „ V3 n_sl3-l % sin2 4 2 2 3" 2 12 2. T a c o : F ' ( x ) - Ii = jxVx + ldx Khi do I i = J(t - 1 ) V t d t = j t V t d t - JVtdt 5ax^+(3b-8a)x + c - 4 b 2 5 3 Cach2: 1^ = j{x + l)yf^dx- , j -1. «i T a l u o n c o : F'(x) = f ( x ) , V x > 2 khi v a c h i k h i 2 = t ^ V t — t V t + C = 2tN/t 5a = 20 a=4 3b-8a = -29 b= l c-4b = 7 c=l l 2(x+ifv;^ = j^^dx t_l^ [3 3 + C = 2(x + l ) V ^ 1 + C. = J(x + l ) 7 ^ d { x + l ) - J V ; m d ( x + l ) 2(x+l)V;m ^ ^ x+1 + C = 2(x + l ) V x + l ^ ^ + 1 ^^ +C 127 Cty TNHH MTV DVVH Khang Vi$t cap »'c giJi 10 chuyen dg 10 digm thi man Toan - Nguygn Phii Khanh = 2(x + l)^yx + l f3x-2 15 ) (-sinx.dx) tanxdx = sinx d x = cosx •'cosx Dat a = cosx=>da = - s i n x d x + C. I Ban doc xem cdch gidi sau c6 dung khong?. f(-sinx.dx) B = -p^ J cosx D a t t = V ^ = * d t = - y : L = d x hay dx = z V ^ d t = 2 t d t . ! , . : 2VX + 1 Khido I i = | ( t 2 - l ) 2 t d t - 2 j ( t 3 - t ) d t = 2 2. Dat / x ^ + 9 = x - t = > x : t2_9 2t dx = 1^+9 2t' 1 ? Vay ]i = A - B = - t a n x + ln cosx + C . +C x2-l +C= rx+1 -1 + C = (x + 1) 2 2 2 4 1 c(4cosx + 5)sinx.dx +c. 2J cos 2t 4t + 5 t^+3t + 2 dt ^ 16 4t^ ^ --1621n X - N / X 2 + 9 -+- t + l_ .3(t + l ) + ( t + 2)^^ f Xi-rt'|fi:J(x)'i|:'- ( t - f l ) ( t + 2) dt = -31n|t + 2|-ln|t + l l + C tanx , f sinx dx 3. J3 = J — 3 - d ' ' = 4 COS x Dat t - COS COSX -dt6561 dt = - = -31n|cosx + 2|-ln|cosx + l | + C. t'* 6561 1 / 3 _ 1 6 2 65_61 l d t = - i - -—1621nt T- + C 1616 4 4t* x-V? +9 3 •vt + 2 ^-i-dt .Dat t = cosx=>dt = - s l n x d x X + 3COSX + 2 Khi do J2 = - .(t^-8lf -t^-9 fda , ^ , — = - l n a+C2 = - i n c o s x + C-2 •' a X => dt = - s i n x d x => sinxdx = - d t 1 -+ C- +C Kr- + CT 3.cos^x 16 Dang 3: T i m nguyen ham bang phu-o-ng phap tirng phan Vi du 2. T i m nguyen ham : 1. =^ [tan^xdx 2.J2 = 5sinx + 2sin2x •dx cos2x + 6cosx + 5 3. h = tanx Cho hai ham so u va v lien tuc tren [a;b] va c6 dao ham lien tuc tren fa;b dt = cos^x dx A = j _ l _ t a n x d x = jtdt = 1 9R Can phai lira chpn u va dv ho-p l i sao cho ta de dang tim du-o-c v va tich phan + Cy - ^ l a n ^ x + vdu de tinh b a n udv . Ta thu-ang gap cat dang sau 129 Ca^p tOc giai 10 ChuySn 66 10 die'm thi mOn Toan - Nguyjn Phu Khanh D a i J ^ i . - I = fP(x) sinx Cty TNHH MTV DVVH Khang Vif t d x , trong do P(x) la da thu-c. cosx sinx Vo-i dang nay, ta dat u = P(x), dv = cosx |sin2x.e^^dx = i e ^ \ s i n 2 x - - j c o s 2 x . e ^ ' ' d x = i e ^ ' ' . s i n 2 x - - l 2 3x dx. e -^e^" cos2x + - e ^ ' ' . s i n 2 x - - I 2 =^ I2 = • ^ ( 3 c o s 2 x + 2sin2x) + C . "13 =>h= Vai dang nay, ta dat du = 2 l ^ d x u = ln x 3. Dat u = P(x) X dv = (2x + l ) d x V = X , trong do P(x) la da thu-c dv = e'*''^''dx + X I3 = ( x ^ + x ) l n 2 x - 2 J(x^ + x ) l n x d x Dang 3: 1= f p ( x ) l n ( m x + n)dx u = ln(mx + n) Vo-i dang nay, ta dat Dang sinx 4: \ cosx Dat: e'dx dvi = ( x + l ) d x d e t i n h fvdu ta dat u = sinx Vay I 3 = f x 2 + x ) l n 2 x - ( x 2 + 2 x ] l n x + — + 2X + C. cosx Dang 4: T i n h tich phan bang phiro-ng phap phan tich Vi du. T i m nguyen ham: 2.U- •' fcosZx.e^^dx; 3. I3 = f(2x + l ) l n ^ x d x dv = dx du = dx v = -cotx • 1^ = - i x c o t x + - [cotxdx 2 2-' sin^x 1 1 ;d(smx) = —xcotx-sinx 2 2 1 — x c o t x — I n sinx + C . 2 2 V I ( s i n x ) ' = cosx = > d ( s i n x ) = (sinx)'dx = cosxdx 2. Dat Dat u = cos2x du = - 2 s i n 2 x d x 1 3x 3 dv = e^''dx u-j^ = s i n 2 x d v i =e^''dx 1. dx A0 2. B = x''+3x + 2 \ ' 1, '2 = - 6^^ " cos2x + - fsin2x.e^''dx. 3 3J J- 5 x - 1 3 dx; x^ - 5 x + 6 _|(;^ + 2 ) - ( x + l)^^^_V 1 dx J(x + l ) ( x + 2) J (x + l ) ( x + 2) = ln I 2. T:ic6•(x-3)(x-2) V 3 x-3 x-2 x-3 x+1 x+2 ^ x-2 0 =-e 3x ^' x ^ + x + 2 0x''-4x +4 x+1 1 x+2 _a(x-3) +b(x-2) (x-3){x-2) fa=3 ^[h dx = (31n|x-2| + 21n|x-3|) = - l n l 8 . dx = dx , 2 , 1 ln--ln- = lnl 3 2 3 d u j =2cos2x Vi 3.C^J^>^^^dx. 0 X -4x + 4 Lo-i giai at' =(ln|x + l | - l n | x + 2|) D e y : A = fcotxdx= ^"^^dx, •' •'sinx [[ V i d u . Tinh tich phan: 1 Lo-i giai 1. Dat ! dv = e''dx dv^e^dx u =x 1 2 = - x +x Khi do J(x + l ) l n x d x = i ( x 2 + 2 x ) l n x - i | ( x + 2)dx = ^(x^ + 2 x ) l n x - x + C' cosx = { ^ dx ; -"l-cosZx dx X Vi sinx ^ du^ = Ui = l n x dv = P{x)dx Vo-i dang nay, ta dat , 5x-2 dx 1 +V X^ - 4 x + 4 = 2, - Cty TNHH MTV DVVH Khang Vigt ca'p ttfc giii 10 chuy6n dg 10 difi'm thi mfln Toan - Nguygn Phii Khanh Ta c6: C= 5x-2 5x-2 x2-4x + 4 5 x-2 -+- 8 (x-lf A ^ B _ A ( x - 2 ) + B _ fA = 5 5^-2 ( x - 2 ) ^ dx = 51n x - 2 (x-2)^ 1 0 x-20 (x-2)2 = 51n- + 4 2 [8 = 8 :• . ' • v T d u 1. Tinh tich phan: 2 -dx 1. I l = 1i + 7 x ^ t 3. I3 = J ,,,,,,„ j ^ ^ , CM:' -3 1 2.l2= J • i ^f. dx xVT^ dx x^dx 4. L = 2x + l + V4x + l ox + Vx^ + 1 Dang 5: T i n h tich phan bang phirang phap doi bien so 1. Dat t = V x - 1 •t^x = t^ + l o d x = 2tdt 1. Phiro-ng phap doi bien so loai 1 . 1 • Doi bien: x = l = > t = 0, x = 2 = > t = l ( Gia su- can tinh I = f (x)dx , ta thirc hien cac birac sau: Vay, I i = ^ — - : - 2 t d t = 2 1 +t Biro-c 1: Dat x = u ( t ) {v&i u ( t ) la ham c6 dgo ham lien tuc tren [a;P], f ( u ( t ) ) ;fdc trer? [a;p] vd u ( a ) = a, u(p) = b ) va xac djnh a, p. Biro-cZrThayvaotichphanbandautaco: ^.^ a a Vay, I2 = Mot SO dang thirang diing phirang phap doi bien so loai 1 Ham so dirai dau tich phan chu-a „Vb^x^ - a ^ ta thu-ang dat x = j2 +2t-2ln|t + l =2 = il_4l„2 3 Doi can: x = - 8 = i . t = 3 , x = - 3 = > t = 2 : * Ham so dirai dau tich phan chu-a Va^-b^x^ ta thu-ang dat x -dt = 2 | t ^ - t + 2 - — dt t+1 0i[ 2. Dat t = N/TOC r : > - 2 t d t - dx 1= J f ( u ( t ) ) . u ' ( t ) d t = Jg{t)dt = G(t) P = G ( P ) - G ( a ) . * ^3 = 2 _ - _ ^ 0 t +1 ^sin t b bsint f * Ham so du-ai dau tich phan chu-a a^ +b^x^ ta thu-ang dat x = ^ t a n t * Ham so du-ai dau tich phan chu-a ^ x ( a - b x ) ta thu-6-ng dat x = -^sin^ t dx i tdt — 7 = d x = -2 7 , i i - t ^ t = ( l n | t - l | - l n | t + l|)^ = In d^_^f(t+l)-(t-l) (t-l)(t+l) i t ^ - i t-1 t+1 = lni-lni =ln3 2 3 1 3. Dat t = V4x + l = > t ^ = 4 x + l ^ d x = i t d t 2 Doi can: x = 2 = > t = 3 , x = 6 = > t = 5 6 Do do: 2. Phu-o-ng phap doi bien so loai 2 Tu-ang tu- nhu- nguyen ham, ta c6 the tinh tich phan bang phirang phap doi bien so (ta goi la loai 2) nhu- sau: b De tinh tich phan 1= j f ( x ) d x , neu f ( x ) = g [ u ( x ) ] . u ' ( x ) , ta c6 the thu-c hiei I3 - s f. tdt - J ^ 2x + 1 + V4X + 1 J|(t In t + 1 + ir 1 t +1 dt ( t + 1)^ =i n l - l 2 12 x-Vx^ + 1 rdt = u'(x)dx. x^ + l - x dx D 6 i c a n x = a=>t = u(a), x = b=>t = u(b) Bu-6'c2:Thayvaotac6 u(b) 1= fg(t)dt = G{t) b a• dt = J x 3 N ^ ^ d x - V d x = 1-^ 0 n = , _ i , vai J = jx^VTTTdx 5 ^ 5 J :UAi Cap toe (jicii 10 clmyen 6i 10 di6'm thi mfln Toan - Nguyln Phu Khanh Dat t = Vx^ + 1 => Cty TNHH MTV DVVH Khang Vi^t + 1 => 2tdt = 2xdx => tdt = xdx = L a i giSi Doi can: x = 0 = > t = l , x = l = > t = 72. u = Inx V2 • > V / J \ 1 .5 3 2V2 2 15 15 ^f3sinx-2cosx Vi d u 2. Tinh ti'ch phan: I = / (x + 1)^ Khido A = - 1 -.Inx , ' X + o ( s i n x + cosx)^ Jt 21 = 1 + 1 = ^x- 0 (sinx + cosx) = 1 — f ^ - 0 2cos' x - 1 ^/x(x + l ) J x(x + l ) -dx = 1 - I 4 , 2 I X — dx dV: v^m ^r3cosx-2sinx , 1 kix = dx o(cosx + s i n x ) 0 (sinx + cosx) 1 =-tan x+1 dx = (ln X - I n x + 1) 1 e ^\n——ln-e_ = lne = l . 1 e+1 1 ^ , e e x+1 u = Inx 2. Dat: X X = -l+I Vay, 1 = 0 . 7C X- dx 1 In ^ f 3 c o s t - 2 s i n t ,. 3cosx-2sinx dx dx = - d t , x = 0==>t = - , x = - = > t = 0. 2 2 2 K1X = V = x+1 Lo-i giSi ^r3sinx-2cosx , -1 / 0 (sinx + cosx) Suy ra: I = X dx dv = - t V 1 4^2 2^21 t f f t ' ^ - t2 ) d t = •(t2-l]t.tdt= Khido 1, Dat dx du = du = [v=2VxTi K h i d o : B = (2Vx + l l n x ' -1 4J Vai 1 = dx -2 dx = 6 I n 8 - 4 1 n 3 - 2 I d x . D a t t = Vx + l 4; Dang 6: Ti'nh tich phan bang phu-ang phap d d i tirng phan Cho hai ham so u va v lien tuc tren [a;b] va c6 dao ham lien tuc tren [a;b Khi do : j u d v = uv V i du 1. Tinh tich phan: vdu =>! = t^-l 2t + ln .2tdt = 2 t-1 t +1 t^-l 1 ^ dt dt = ^^2 + . ^ t-1 t+lj = 2 + ln3-ln2 Vay, B = 2 0 1 n 2 - 6 1 n 3 - 4 3. C = J — 1 dx = J — ^ - A i x + j x V 5 ^ . d x = K + H e u = ln(5-x) „ 1.1n(5-x) K= J — L — ^ x . D a t • ^ dx dv = — x du = - dx 5-x v--i X 135 Ca'p tO'c giai 10 chuy6n dg 10 di6'm thi mfln Toan - K= - ln(5-x) - ^ ^ Cty TNHH MTV DWH Khang Vi$t Nguygn Phu Khanh = ln4--f-ln(5-x) lf(x)|dx = •f(x)dx + lnx^ =-ln4 khoang ( a ; b ) . 4- H = jxyjs^.dx.Dat t = y j s ^ =>2tdt = - d x T;^ Doican: x = l = > t = 2, x = 4 ^ t = l b b Neu: f (x) > 0 , Vx 6 [ a ; b] t h i | f (x)|dx = j f {x)dx a a b b Neu f ( x ) < 0 , V x e [ a ; b] t h i |f(x)|dx = - J f ( x ) d x ,..D 164 1 H= j(5-t2)t(-2t)dt =2 3 Chiiy: Neu phu-o-ng trinh f ( x ) = 0 c6 k nghiem phan biet Xi,X2,...,X|^ tren (a; b] / Liri gi&i dx Dat u = ln(x + 2 ) , d v = - p = = d x . K h i d 6 du = — - ,v X V4-x^ b S = | f ( x ) | d x = j f ( x ) d x + Jf(x3dx + ...+ | f ( x ) d x a = -V4-V ^'^^ Cong thu-c tinh dien tich hinh phang gi6i han bo-i cac du-6-ng: Theo cong thu-c tich phan tirng phan, ta c6 -1 + °-V4^ dx = -21n2 + -1 x+2 -1 x+2 b y = f ( x ) va y = g(x) va hai du-6-ng thang x=:a,x = b(a < b ) : S = J f ( x ) - g ( x ) d x . a dx. Vi du 1, Tinh dien tich S ciia hinh phang H gio-i han bo-i: Dat x = 2sint. Khi do dx = 2costdt. Doi can: x = - l = > t = - - , x = 0=:i>t = 0. 6 X+ 2 dx = ' Acosh 2sint + 2 0 dt = 2 ( l - s i n t ) d t = 2(t + cost) = 2+ ^-7^. 1. Do thi ham so: v , true hoanh va du-o-ng thang y = 2 - x . r i l ' 2. Do thj ham so: y = (e + l ) x va y = (e" + l ) x . 3 Lo-i gi^i 1. Hoanh do giao diem cua do t h i ham so: y = -\fx Suyra I = - 2 i n 2 + 2 - 7 3 + j . Dang 7: IJng dung tich phan Bki toan 1: Dien tich hinh phang gio-i han va du-6-ng t h i n g y = 2 - x la nghiem cua phu-o-ng t r i n h : - V x = 2 - x<=>\/x = x - 2 x>2 x>2 x = x^-4x + 4 <=> [x^-5x + 4 = 0 2 <=>x = 4 -yirn ik'i {,,,,3; 4 Dien tich h i n h phang H : S = Vxdx + J^2 - x + Vx jdx Cho ham so y = f (x) lien tuc tren [a;b]. Khi do dien tich S ciia hinh phang (D) 2 0 2 gio-i han bai: Do thj ham so y = f ( x ) ; true Ox : ( y = 0 ) va hai du-crng thang ^7x3 x = a;x = b la: S= | f ( x ) d x . 7,i| Khi do tich phan S = J f { x ) dx du-o-c tinh nhu- sau; ' dx. V4-x2 -1 0 khongdoiddu. thi tren moi khoang (a;xi),(xi;x2)...(xi^;b) bieu thu-c f ( x ) °fXln{x + 2)^ Vi du 2. Tinh tich phan: I = I = - V 4 - x ^ l n ( x + 2) , 15 5 Vay, C = K + H = ^ l n 4 + ^ 5 15 1= cong thu-c nay chi diing k h i f(x) khong doi dau tren a .3 0 + n 2x ^ 2/3 ^ 2 + 3-Vx^ 2 = 4V2 ' (16 +[ 3 2 4V2^ 3 10 ," 3 2. Phu-o-ng t r i n h hoanh do giao diem: 137 Cty TNHH MTV DVVH Khang Vigt cap t6"c giai 10 chuySn dg 10 die'm thi mOn Todn - NguySn Phu Khanh (e + l ) x = ( l + e'')x <=>x(e''-e) = 0 <=> x=0 x=0 A = (m + l f - 4 m > 0 e''=e x=l ' in + 1 > 0 • •••'^ m>o 1 , <••> 0 < m A 1 : . Dien tich hinh phang H : S = J ( e + l ) x - ( l + e ' ' ) x d x = J x ( e - e ' ' ) d x Vo-i 0 < m : ? i l t h i phu-cng t r i n h (2) c6 2 nghiem la t = l , t = m , v i m > l nen Vai V x € [ 0 ; l ] , t a l u 6 n c 6 : 4 nghiem phan biet cua ( l ) theo thir tir tang la: - \ / m " , - 1 , 1 , x(e-e'')>0 VnT Theo bai toan, ta c6: dx Vay, S = 1 ^Hl ^ ^ " 2 0 / \. dv = ( e - e ' ' jdx 1 S= V 1 j ^ ^ - ( m + l ) x ^ + m dx = x'* - ( m + l ) x ^ +m|dx 0 du = dx U = X Dat Vm = ex - e o e ^ j d x ^ - ex 2 2 x ' * - ( m + l ) x ^ + m dx = - x'^ - ( m + l ) x ^ + m dx 1 0 e — e -(-1) ^^-13. 2 ^ - ( m + 1) x + m dx = 0 ^ - ( m + l ) — + mx 0 ln8 + ldx. In3 Dat t = V ? T l < = > t ^ = 6 " + l = > e ' ' = t ^ - l = > e ' ' d x = 2tdt hay dx = 2t dt t^-l Khido: S= ^ 2t2 [-F-dt- .^t^-l t-1 f 2 + - ^ dt = 2t + ln t+1 A t ^ - l j = 2 + ln .2, Vi du 2. = 0 0 •(! ' m m+l ^ „ ^ <=>+ l = 0<=>m = 5 5 3 Vay, m = 5 thoa bai toan. 2. Do thj ham so cat Ox tai 4 diem phan biet <::> x''^ - ^m^ + 2Jx^ + m^ + 1 = 0 (*) hay ^x^ - 1 j^x^ - m ^ - 1 j = 0 c6 4 nghiem phan biet, tire m^O . Vai m^O t h i phu-ang t r i n h (*) c6 4 nghiem phan biet ± 1 ; ± Vm^ + 1 1. Cho ham so y = x ' ^ - ( m + l ) x 2 + m c6 do t h i ( C ^ ) . X a c d i n h m > l de do thi (C^) cat true Ox tai 4 diem phan biet sao cho hinh phang giai han bai Dien tieh phan hinh phang giai han bcci (C^) v a i true hoanh phan phia tren true hoanh la: (C^) va true Ox c6 dien tich phan phia tren true Ox bang dien tieh phan 1 S = 2 j x ' * - ( m ^ + 2 ) x ^ + m^ + l dx = phia du'o-i true Ox . 2. T i m cac gia t r i tham so m e R sao cho: y = x"*^ - |m^ + 2jx^ + m^ + 1 , c6 do thj (C^) cat true hoanh tai 4 diem phan biet sao cho h i n h phang giai han bcci (C^) v a i true hoanh phan phia tren Ox c6 dien tich bang 96 15 Lcri giSi 1. Do t h i ham so cat Ox tai 4 diem phan biet o x ' ^ - ( m + l ) x ^ + m = 0 ( l ) c6 4 nghiem phan biet <=> t ^ - ( m + l ) t + m = 0 (2) c6 2 nghiem du-ang phan biet 96 20m^ + 16 96 15 B^i to^n 2: The tich vat the tron xoay Tinh the tich vat the t r o n xoay khi quay mien D du-ac giai han b a i cac du-ang y = f ( x ) ; y = 0;x = a;x = b quanh true Ox . Thiet dien eiia khoi t r o n xoay cat bcci mat phang vuong goc v a i Ox tai diem c6 b X hoanh do bang x la mot hinh t r o n c6 ban ki'nh R=f(x) nen dien tich thiet dien bang 139 cap tOc ylal 10 chuyen d6 10 diem thi mOn Toan - NguySn Phu Khinh Cty TNHH MTV DVVH Khang Vift S(x) = TTR^ = Tif^ ( x ) . Vay the ti'ch khoi t r o n xoay du-gc t i n h theo cong thu-c: b Dat t = t a n x = > d t = b V = Js(x)dx = 7 t | f ^ ( x ) d x . p 1 dx cos^x Khi do l2 = J(l + t ^ j d t = t + - t ^ + C = tanx + i t a n ^ x + C. Vi du. Cho hinh phang H gioi han boi cac diro-ng : y = x l n x , y = 0, x = e. Tinh I3 = J — l - ^ X = j - Lo-i giki • " l + smx •' Phu-ong t r i n h hoanh do giao diem: x i n x = 0 = > x = l . du = . |'u = ln2 at: I Dat: dv = x^dx X 1 2 — In X V = fx^lnxdx^ u = lnx I n v a i 1,= fx^lnxdx 1 (e' 3 9 3 2 2e^+l (5e^-2) 3 "^7 V 9 l] I, x' 14 2, U 2) =I- cosx , dx I3 sin^x-5sinx + 6 +3. = (•V21nx J dx 1 AH i. -dx - ^ l - s i n ^ x j ssin^x i 'cosxsin'x = lU>' 9 y I (dvtt). dx dt- 1 1 t 2 ,t-l In 1 dt = t+1 t t'-i) t+1 sinx 2 '0^ 1, (lnt-l-lnt+l )+C 2^ ^ sinx-1 t-1 dt sinx + 1 +c. V -dx sin^x-5sinx + 6 Dat t = s i n x = > d t = cosx.dx. f i r ^ J ^ ^ i ^ t-3 Hxr&ng dan giai >3 = J t...«!. 1 _ f( (t t- -22) -) (- t( -t 3- 3 )^^ ' J(t-2)(t-3)'^*=J{t-2)(t-3) .1 -v, "•'• —-^dt = ln|t-3|-ln|t-2| + C -dt- V21nx + 3 f 1 f/ 9 \ '2=1—T~^^= 1 + tan^x . dx cos X ^ ' ' cos^x '"'^ Kt' cosx I3 = f — 1 — d x • " l + sinx = Jtan^xdx = J — ^ — 1 d x = — ^ d x - J d x = t a n x - x + C. cos X" 'it't ^^^^ ^ ^(i-t'y (i-t^jt^ 2e^+l 1. Tinh nguyen ham 140 = -tan (n Dat t = sinx => dt = cosxdx B^i tap t y luyen s 2; 2 2 cos cosx f1 1 tan^xdx d ^ d- d xx cosxsm x V = - 9 x^ J dx "x3" 3 2) 2, j I1 = h dv = x dx ^4 v4 2) 4 Hiro-ng dan giki 1 du = x) X^ dx = — X -Inx = •l = . — U 2cos' 1^71 2, Tinh nguyen ham 1 *• 1^ Vay, Vox = ^ tan 0, 1 Dat: 2inx_, dx - * ^ ^'"'"^ ' 4 Vay, V o , = 7 t ] ( x l n x f d x = T t ] x 2 l n 2 x d x = 7cIi 7t _ X ' -2d the tich cua khoi t r o n xoay tao thanh khi quay hinh H quanh true Ox . dx = l n t-3 t-2 + C = in sinx-3 + C. sinx-2 2 1 1 Dat t = 21nx + 3 = > d t = - d x = > - d t = - d x . x 2 x el r 12 r ^ tv/t ^ {21nx + 3)x/21nx + 3 Khi do 13= f i V t d t = - . - t V t + C = ^ + C = -i ^ J2 23 3 L 3 +c 141 Cty TNHH MTV DVVH Khang Vi§t cap ta'c gi^ii 10 chuyfin dg 10 die'm thi man Join - NguySn Phu Khanh Doi bien: x = 2 = > t = 3 , x = 2^5 => t = 5 3. Tinh tich phan: 1 a. 4x-2 1= J(x2 + l)(x + 2) a. Ta c6: b. J = dx 4x-2 _ (x + 2)(x2 + l ) A ^. ^-x^dx Khi do: I = x^-9 0 Hiro-ng dSn gi^i ^ Bx + C ,.2/ x^(A + B) + x(2B + C) + 2C + A x+2 x^ + l Dong nhat thirc 2 ve, ta diro-c: A = -2 2B + C = 4 B=2 2C + A = 0 C=0 21n x + 2 + l n x ^ + 1 0 t' , Khidol3= 1 J . ^ 1 _ 1f d. U = M= dt ^ ^n--lnl 18 2 ^ 1 t+3j dt 0 x-3 272 3/ 3 r V x - x + 2011X j X = - .dx = 2t t^-9 2t2 dt Doican: x = 3 = > t = 3 x = 5 = > ! ; - 9 Hay I i = - 3 + 6 1 n Khi do 1 = b. Dat t = Vx^+5 => t^ = x^ + 5 =>xdx = tdt 2 14077 2 1 ^ 1+ a. Dat t = ^/x + l = > t ^ - l = x = > 2 t d t - d x x = 3=>t = 2 dt 2 1 ^ J2011x-^dx^- H i r a n g dan gi&i Doican: x = 0=i>t = l , 2^ —dx+ I 2J2 N= ay, I4 = M + N = 3 ^ t j ln--ln4 2 2^/2 3 - T - l dx= X ^ , d-l4= t-1 1^ dx.Dat t = 3|-l--l = : i > t ^ = - l - i ^ 3 t 2 d t : = - A d x i 3 1, 1 —In18 2 xdx 3.Vx + l + x + 3 1, 15 = —In — 4 7 t+2 1 Khi do M = „ 4 Doi can: x = l o t = 0, x = 2V2 => t = - S. Tinh tich phan: 3 dt l1.t-(t-l), iV 1 p ^= -^dt=2Jt(t-l) 2J t ( t - l ) 2JU-I 4 - - 2 1 n 3 + In2 + I n 2 - l n l = l n 9 t-3 = — ( l n t - 3 l - l n t + 3)^ - l l n 18^ ' 0 18 t+3 t+2/ I :'(;iI) Si _ J l_ \%{(tt + 33))--{ (tt- -33) ^ ^ ^ 1 V 1 18 t-3 ^ ^ 3 J t 2 - 9 = 3 J ( t - 3 ) ( t - f 3 ) = 1 8 J ( t - 3 ) ( t + 3) f 4JU-2 = -(ln|t-l|-ln|t|) = - l n 2 2 b. Dat t = x^ = > d t = 3 x ^ d x -1 3Jt2-4 t-2 Doi can: x = l = > t = 2, x = N/^=>t = 4. 4x-2 J V 2 2x -dx= +dx Vay, 1 = o ( x 2 + l ) ( x + 2) oV ^ + ^ ^^^ + 1 1 1^ c, Dat t = l + x =>dt = 2xdx. (x + 2)(x^ + l ) A + B= 0 t2_4)t ^. rlt- t'-9 2t^ kit=2 j t - - tj dt = - 2 --9In = 18—in3 2 ,