C-UON TRAN BA HA AM NANG LUYI^THI DAI HOC IIGUYEN HOM * * THU VIENTINHBINHTHUAfO mk nXr ikn i « i nc wfc cu M Mi Cty TNHH MTV D VVH Khang Vi^t LOI N O I £>AU Phanli I N G U Y E N H A M - T I C H P H A N V A \SNG Chuong 1: r .•,, NGUYEN HAM Tap sdch gom phan: A T O M T A T LY THUYET Phan I: Nguyen ham-Tich phan vaung dung D i n h nghia: Cho ham soy = f(x) lien tuc tren khoang D D U N G ^, ' F(x) la nguyen ham cua f(x) tren D va chi khi: F '(x) = f(x) Vx e D Phan 11: So phuc Moi phan diroc trinh bay theo tung chirong, moi chuong bao gom cac chuyen de, moi chuyen de du-oc phan cac van de co ban, moi van de bao gom: Tom t5t kien thurc - phuong phap giai - bai tap ap dung - bai tap tu luyen Cuoi moi chvrong deu co phan Bai tap tong hop va Bai tap luyen thi bao gom cac bai tap nang cao duoc tuyen chpn qua cac de thi dai hoc va cac de thi hoc sinh gioi Hi vong rang tap sach co the giup ich cho hoc sinh cac ki thi hoc sinh gioi, ki thi dai hoc Rat mong s u gop y cua doc gia va dong nghiep de Ian xua't ban sau tot hon Tinh chat co ban: + Neu F(x) la mot nguyen ham cua f(x) tren D thi F(x) + C cixng la nguyen ham ciia f(x) tren D (C la hang so) + Neu F(x) va G(x) la cac nguyen ham cua ham so f(x) tren D thi ton tai hSng soCdeG(x) = F(x) + C + Ky hieu: jf(x)dx = F(x) + C ( l a h o nguyen ham ciia ham so f(x)) + Neu f(x) va g(x) co nguyen ham tren D thi: l[f(x) + g(x)dx = Jf(x)dx + l(x)dx jkf(x)dx = kjf(x)dx, ke R Trdn Bd Ha Gido vien THPT Chuyen Le Quy Don - Dd Ndng Tu nghiep tgi: lustitut de Recherche Pour L 'enseignement des Mathe 'matiques Paris-France Nha sach Khang Viet xin tran giai thieu tai Quy doc gia va xin long nghe moi y kien dong gop, decuon sdch cang hay han, botch han Thuxinguive: Cty T N H H Mpt Thanh Vien - Dich vu Van hoa Khang Vi?t 71, Dinh Tien Hoang, P Dakao, Quan 1, TP H C M Tel: (08) 39115694 - 39111969 - 39111968 - 39105797 - Fax: (08) 39110880 Hoac Email: khangvietbookstore@yahoo.com.vn + Neu Jf(x)dx = F(x) + C thi Jf(ax + b)dx = - F(ax + b) + C a + Moi ham so lien tuc tren D deu co nguyen ham trenD Y0 Bai 8: Tim a, b, c de F(x) = (ax^ + bx + c)e^ la mot nguyen ham cua f(x) = x V Giai e^-l-x (x-l)e''+l ,- 2x e'_l 2 x ^ O = f(x) r:>dpcm Vay: F "(x) = X = 'V " ' De F(x) la nguyen ham ciia f(x), Vx e R thi phai c6 Bai 11: Chung minh Vx ^ thi Vn e N', ta c6: a =l F '(X) = f(x) Vx € R r:> b = - b +c=0 n.x""'-(n + l)x°+l ^ c =2 = + 2x + 3x^ + + n.x-^' (x-1)' Giai Vay: F(x) = (x^ - 2x + 2)e« Bai 9: Tinh dao ham ciia ham so': F(x) = (x^ - 1) In + x - x^In Xet ham so' f(x) = + 2x + Sx^ + + n x " ' 1-x" Tu suy nguyen ham ciia ham so': f(x) = xln Ta c6: F '(x) = 2xln 11 + x , , v6i X 9t va X 7t Ta c6: f(x) c6 nguyen ham la F(x) = x + x^ + x ' + + x" = x Giai Theo dinh nghia ta c6: F '(x) = f(x), Vx ^ -(2xln|x| + f r ) 1-x" « f(x) = x.1-x x+1 n.x"-'-(n + l ) x " + l 1-x > dpcm (x-1)' = 2xln(l + x) - 2xln I x | + (x - 1) - x Vay F '(x) = xln 1+x Bai 12: Cho f(x) = xln - va g(x) = x^ln 4 -1 Xet G(x) = F(x) + X => G '(x) = F '(x) + = f(x) nen G(x) la nguyen ham ciia f(x; Vay: G(x) = (x^ - l ) l n I x + x | - x^ln I x I + x la mot nguyen ham ciia f(x) Bai 10: Chung minh F(x) = khix;t0 X = Chung minh: f(x) = g'(x) - ^ x tir suy ra: Jx.ln ^dx Giai g(x) = x^tn - => g"(x) = 2xln - + x^ - = 2xln - + x 4 x ( x - l ) e ^ +1 x ^ Do do: - g'(x) - - x = xln - = f(x) ^ x = f X X~ X Suy ra: fx.ln —dx = — g(x) - — + C = — x^ln J 4 la nguyen ham ciia f(x) = X +C K^uiii iiuii^ luy^n ini uii - ivguyeri nam - i icnpmm - AOpmtC - Iran tsa na Vamde2:TIM HQ NGUYEN HAM CUA HAM SO y = f(x) BANG DINH NGHIA Phuong phap: Phan rich f(x) tong (hieu) cua cac ham so' ca ban c6 the ^> 2x + l tim nguyen ham bang each ap dung bang cong thuc nguyen ham ca ban, ap 4x'+4x l dung rinh cha't cua nguyen ham de tinh hoac dua ve dang nguyen ham cua 1 2x^ x" 1 f(2x-+x -—^dx=—+^-TX lr|2x+l|+C 2 2(2x-l) dx= 2x+l ham so'hgp dx dx r Bai 1: Tim hp nguyen ham cua f(x) = cosxcos3x 2(2x-l) fV3^-V3x-2 Giai S jx-dx - - J(3x -1)-' d(3x - 2) = - X V ^ - -(3x - 2)V3x - + C f(x) = ^ [cos4x + cos2x] Jf(x)dx = — J(cos4x + cos2x)dx = — [ — sin4x + — sin2x] + C 2 Bai 2: Tun hp nguyen ham aia f(x) = 2'"' -5 x-l -dx = f2.2" x-1 - du = 2xdx, do: i-'• • '•••tdu = -I du J= b) Dat X = asint-— dx = acostdt ,u - In J= fVa' - x" , f a cos t.a cos tdl —dx= : J Y a sin rcos" t , ^ dt t ff, > du f(x)dx = a = a + -^ u'-l \ a" - X' V 2(ln3-ln2) 3^-h2'' a) fcos X , f(l - s i n - x ) < I dx = \ ^cosxdx = sin X sin X V sin 1 b) f ' dx = f (sin x + COS x ) " • ' l + sin2x J^'^i ' , do sin Va" - X' -a dx •' ' sin - A Va" + X" +a Bai8: 4j Tinh j(x-l)e^"'-"''"'dx Giai Dat u = x2 - 2x + => u' = 2(x - ) (x - l ) e ' ' ' - ' " M x = i je-u'dx - i Je-du Dat u = (1 + xe") => du = (x + l)e''dx, do: U=COt2xr:>u' = - = -2(1 +cot22x) sin" 2x du = In u _x-"-2 Giai u -1 71 X+ — x+ 8419: Tinh: J(l + cot^Zx) e-'^-dx u(u-l) —cot + C -dx xO + xe") rf d(sin x ) :-tk 4J x +1 du X X sin^x + C < " " ^ " -dx xe^(l + xe^) r + C + sin2x Giai 1= 3^-2' -In dx vi cost = V l - s i n - t = j l - — = + ^ In -i-C = b)f(x) = + C = a cost + — I n cost + X" u-Hl = In sinx | cos t - f (x)dx - V a ' - 2(ln3-ln2) u-1 Giai 1, du = a u + - l n + C u +1 u+1 -In a) f(x) = — sinx u-du = a 1+ — u-1 V du Bai 7: Tim ho nguyen ham cua cac ham so': J sin t Dat u = cost => du = -sintdt / u -^ 1^ V a ' - x" = aVcos" t = a(cot) = acost 1= In—dx "du = u + C = In I {In(lnx)) I + C xlnx.ln(lnx) Bai 5: Tinh I = u = u-1 + C = In xe^ + xe^ + C J(l + cot22x) e-"'2' z" = z''(cos27i + isin27i) = 8^2 - c o s — i s i n — b) 2" = 8N/2(- — - l i )= - > / - V i B a i 2: V i e t d a n g d a i so c u a so phi'rc: z = •K n cos — i s i n 12 12 1-i 7t 71 COS 12 sin 12, = Sin " ^ ^, 7t D o d o COS 12 4i 7t 71 7t 71 1+ i s i n ( 57t cos ^/6-^/2 71 N/2 + >/6 isin — = 12 371 cos y V 16 16* r- i ( V + V - ( V - V ) i ) a) b) ll-2iV3j Giai = 27t 27:^ c o s — + isin — 3 21 + i 271 cos 4i z = a + b i t h i a = 6cos — = - 3 =3N/3 dod6:z =-3 +3N/3i = -64-64i N21 l-2iV3 V3)2]=-V2 Giai ^571^ + 3V3i ) (N/3 +1 + N/3 - l ) i ) ( V + - ( V - l ) i ) + isin 371 ^ Bai_2: V i e t d a n g dai so ciia so p h u c N/6-N/2 r 271 27:1 B a i 3: V i e t v e d a n g d a i so so p h i i c z = c o s — + i s i n — 330 J _I_ + i = \/2 (cos— + isin — ) 4^ = cos— O S— —+ sin—sin —= —C COS 4 = - — [ ( + V3)2 + ( l - b =6 s i n — 771 7 C O S — + isin — 4 a-iF=(V2) c) %/3 - i = ( c o s ( - - ) + i s i n ( - - ) Vay z = - ^ ( > / + + ( V - l ) i ) z= — • J l - V i ) ( l i ) _ l ^ ^ ^ ^ ^ _ ,.1-J3 cos ; 1-i Giai CO = 64 b)l - u= V2(cos( ) + isin( ) 4 (i-V3) Ta +if Giai Giai 7TI 7ii c o s — +isin — 6 MTV D VVH KhaiiR isin 71 , +i 271 21 — = 71 sin — j 2012 x2012 COS — + , COS\ 2012 2012 -Ti + i s i n - _ _21()06 ) Vie I Cty TNHH MTV D VVH Khang VK-, = cos(-7r) + isin(-rr) Do CO hai can bac hai cua z la: z Vain de L/NG DUNG CUA CONG THL/C MOIVRE Phuong phap: Co the diing cong thuc Moivre de tinh luy thua bac n, can bac n, chiing minh cac cong thuc lup'ng giac Bai 3: Cho w = Bai 1: a) Cho z = coscp + isincp Chung minh voi mgi so'ty nhien n > 1 Taco: z " + — = 2cosn(t); z " = 2isinn(t) Z2 = -(cos(-^) + i s i n ( - | ) ) = z , = cos(-^) + isin(-^) = - i N/S +i a) Viet can bac hai cua w dang lugng giac b) Chung mmh cos— = ' va sin— = 12 — 12 Giai b) Chung minh: cos> = - (cos4(p + 4cos2(p + 3) v/3 a)w = - V + i = 2 sin> = — (sinScp - SsinScp + lOsintp) 571 z" b) Goi z b) Vi z + - = 2coscp, z - - = 2isincp suy ra: cos''{p = = X 1^ 12 57l' cos w, sin^cp • ' 1^ z — z 4x^+473x^-1 = Bai 3: Tim cac can bac hai cua so' phuc: z = cos / 7t cos I T ) N 71 ' + isin I N/3-1 57t 71 COS 73 + z,= — - — + — ^ — 12 - V3-I y=—r— Vay CO hai can bac hai (dang dai so) cua w la: 72 cos— = -|3 + isin — • Ta c6: z = 7^-1 , 73 + 73 + Dong nhat voi dang lucj-ng giac (Phan a), ta c6: Giai \ hay 1^ z — 1 = — (2sin5cp - 2.5sin3cp + 20sincp) = — (sinScp - 5sin3cp + lOsincp) 16 ( ta c6: x = -i74-2V3 ' 2^i ; + isin 12; O = -(cos 4(t) + cos 2(t) + 3) n5 12 + ism 12 ^ = 2;: 1^ 1 + =—(2cos4(t) + 8cos2(t> + 6) Z J J ^6 z +— z 5n 57t cos— + ism — + iy la c3n bac hai cua z ^ C O S — + ism — 12 12 Do do: z" + — = Zcosncp va z" - — - 2isinncp n4 5n 57t = cos— + isin — 57t 5^t^ cos— + ism — 6 a) Ta c6: z" = cosncp + isincp; — = cosncp - isinncp z - Vay CO hai can bac hai cua w la: Giii z" + 71 ^ 73-1 _57r 76-\/2 -=> cos-12 rr 5n 73 + Sn yfe v2sin—= • -— = • •=>• sin— 12 12 + y/l I- Cam nanf^ luyen thi DH - Nf^uyen ham - Tich phdn - SV; phi'cc Bai 4: Tim n de w = 3-V3i n+1 sm a na na cos — + ism — =P 2 sin la so thuc, so ao? V3-3i Giai ^ , S-Si sis \ n n Ta co: —;= = — + — i = cos—+ isin — V3-3i 2 6 n+1 sin a n+1 na na sin a/ cos—+ isin — (cosb + isinb) a 2 sin na + b na + b + isin sin Do w = cosn — + isinn — n+1 sin a na + b Tu ta co: S = cos a sin w la so thuc sinn — = O c * n — =kTCn = k ( k N) cos Ra Ha Iri'iti w la so ao cosn - = o n — = — +k7in = 6k + ( k e N ) 6 ' n+1 sin a VaT = sin sin Bai 5: a) Cho cac so thuc a, b cho sin ^ ^0 voi moi so nguyen n > 1, xet cac tong: na S = cosb + cos(a + b) + cos(2a + b) + + cos(na + b) JAI TAP TV" LUYEN T = sinb + sin(a + b) + cos(2a + b) + + sin(na + b) pai 1: Viet dang dai so va dang lugng giac ciia cac so phuc sau: Hay tinh: S = IT tu suy S va T b) Chung minh: voi moi so a ^ krc (k € Z) va so nguyen n > 1, ta co: sina + sin3a + + sin(2n - l)a = sin na sin a l)z,= ^ i\ 11 + Tii J li 2; Viet dang dai so cua so phuc: z = va cosa + cosSa + + cos(2n - l)a = sin2na sin a Jai_3: Tim n de so phuc Giai Dat a = cosa + isina va p = cosb + isinb thi S + iT = cosb + isinb + cos(a + b) + isin(a + b) + cos(2a + b) + isin(2a + b) 571 2) z,= tan — + ^ + i ^ V - i ^Vs-i Vs + i V3 + i A" la so thuc 4-3iJ i j : Cho z la so thuc thoa man z + - = Hay xac djnh so phuc w = z + ,2000 + + cos(na + b) + isin(na + b) = p + Pa + Pa^ + + Pa" , = pn + a + a^ + + a") = p „ l - c o s ( n + l ) a - i s i n ( n + l)a =p 1-a n+1 n+1 n+1 sin a sin a-icos a 2 =P a a a sin - sin - i c o s 2 n +1 sin a / n+1 n+1 sm a -icos a sin 334 — jai 5: Chung minh rang cos ^ - cos ^ + cos 1-cosa-isina Hl/ONG DAN GIAI Zl = a a sin —+ icos — 2) ' -l6 S - + —1 2* lb cos ( 3j + isin n I 3jJ ^2 ) 2''[cos(-2a) + isin-2a] = 2* 335 Cty TNHH MTV DVVH 2) Z j = t a n — + i = 571 cos' -1 371 cos 3n sm COS 37t •- STT Bai 5: Xet p h u o n g t r i n h : s i n — + icos — 8 cos7(p + isin7(p = - 3K^ icos— r 77c> I + isin 8; (N/S + I) BaL2:z = Cam nang luyC'ii thi DH - Nguyen ham - Tich phdn - So phiic - Tran Bd Ha COS 37t liTt Sin llTt ICOS rr f 771^ I Zj = cos ,) 57t 971 z- = cos- l " +(V3-i) 7C + isin 4j 71 I ' 4jj { 1171^ cos ll7t + isin ' Bai3:lli 4-3i = i^±iMliil = l i = 25 D o d o : z la so t h y c o Bai 4; z + - = l < » z - z ^ cos — + sin — 4j sinn — = = > n — = k a o n = k ( k e 4 + l = 0=> Z2=- z,= ^r+-i l + 73i \/3 i = c o s ( - ^ ) + i s i n ( - - ) => xf^ ,2(X)0 = cos(-2000-) + isin(-2000-) cos(-2000^)-isin(-2000|) ''I zr+7i^=2cos(-2000|)=2 -v/i 7t 71 Z j = — + — = cos( —) + isin — z^°°° + - ,2a)o L - = 2cos(20001) = - 336 N) cos r —27:^ I \ J/ = -1 sin7(t) = 371 / 37t 771 z, = c o s — + isin — ^ 7 771 z^ = cos — + ism — 7 llTt z, = cos " llTl + isin • ' Zy = ' COS 137t + = c6 tong cac nghiem bang nen 371 71 hay: cos ^ -32(1+ i) cos7(t> = - l 1171 137r + cos + + COS - X =0 t 371 57t De y cos(7: - x) = -cosx nen: cos — + cos — + cos — = — ^ 7 , — = 32N/2 + = 0, p h u o n g t r i n h c6 n g h i e m thoa m a n : 971 + isin- Phuong trinh =0-0" 371 + isin- cos— + C O S — 7 cos 71 z, = cos— t I'^m— ; 7 8 f COS 571 KhangTT^t 27t 37t cos— +cos— = 7 137t + isin Cty TNHH Chuyen dei: B A IT A PT N G H O P MI \ Vi^t S O P H U ' C ''dm nang luy^n thi DH - Nguyen Ta c6: + i = N/2 (cos — + isin —) Bai Tap Tong Hgp B a i l : Trong mat phSng toa (Oxy), tim tap liop diem bieu dien cac so phiic z 4' z = a + bi=>z-i = a + (b-l)i T= 72 (1 + i)z = (1 + i)(a + bi) = a - b + (a + b)i yja^+ih-lf a2 + b2 - 2b + = 2(a2 + b^) « = sjia-hf 2001 •2011 = 7^ cos 2011^ , •Z.-3 a^ + (b + 1)^ = Bai 2: Tim so phuc z thoa man: I z | = va zMa so thuan ao / z = x + iy x^ +y^ - + 6iy z+3 (x + 3)^+y^ x^ +y^ - z^ la so thuan ao va I z I =2 nen ta c6: Z= la^=\l 3^=1 R(cos(l) +isinc))): Bai 3: Tim diem bieu dien cua so'phuc z thoa man: 2< | z - l + i | b = -1 => zo = - i Suy phuong trinh tuong duong (z + i)(z3-27) = » z'= x^ - 3xy2 + (3x2y - y3)i x^-3xy^=18 = Rcos Vay tap hop cac diem M la cung tron tam 1(0; -3 ), ban kinh R = voi y < DSt z = a + bi => z -1 + 2i = (a - 1) + (b + z)i 338 x^+y^-9 6y Vay CO so' phuc: z i = + i; Z2 = -1 - i; Z3 = - i va Z4 = -1 + i z-^ = 18 + 260i (x + 3)^+y^ b = +l < 12 -1 + 2i I < « < (a - 1)2 + (b + 2)2 < Giai z-3 z2 = a2 - b^ + 2abi b^=l , V ^ CO mot acgumen la ( — ) z+3 ^ ^6 Dat Giai Va^+b^=2 cos 5: Tim tap hop cac diem M mat phSng bieu dien so phuc z cho +{a + bf Tap hop cac diem bieu dien la duong tron tam 1(0, -1), ban kinh R = a^-b^=0 ' ' Do T = (1 + i)2'"o + (1 - i)2ooo Giai Vi I z - i I = 11 + iz I » - i= V2 (cos( ) + i s i n ( - - ) thoa man: I z - i I = I (1 + i)z I DSt z = a+bi ham - Tick phan - S6 phitc ~ Tran Bd Ha z =-i ( z - X z + z + 9) = z2 + 3z + = 0z = - ± i , '.dOr.- Vay phuong trinh c6 nghiem: z = - i ; z = 3; z = -3 ±3 \/3 i L7: Tim so phuc b de phuong tinh: + bz + 3i = c6 tong binh phuong hai nghif m bang 339 CtyTNHH MTV D VVH Khang Vie, Giai Giai GQi zi, Z2 la nghi^m cua phuong trinh theo Vi-et ta c6: z, + Z j = -b; z , Z = i z^ + z^ =8 » b2 = (3 + i)2 b2 - i = o (zi + Z2f - 2ziZ2 = » b2 = + i o b = ±(3 + i) | z + l - i | = | z + + i | v a L ^' la so thuan ao z +i 71 COS I z + - 211 = I r + + i I o (x + 1)2 + (y - 2)2 = (x + 3)2 + (4 - y)2 z= Tap hgp cac diem M(x, y) la duong thang y = x + _ z - i x + (y-2)i x - ( y - ) ( y - l > f x(2y-3)i w — -= — — z+i x+(l-y)i x^+(l-y)^ w la so thuan ao => x^ - (y - 2)(y - 1) = va: x(2y -3)^ 3>/3 71 371 ' 371 + ism + ism l-i 12 l-i do: -=a+i) — + — a) z2 + z = b) I z - 2z = -1 - Si c)x*-2x2-3 = d) z z - ( z + z ) = 3i -19 e)z + 2z =(1 +5i)2 Giai ^ ~- + Z|Z; ) z = a + bi =0 z2 = a2 - b2 + 2abi; z = a - bi Phuong trinh tro thanh: a2-b2 Giai b = 0=>a = Ova = - l : z = hay z = -1 l'-2 b = 2a => a2 + a - Z,Z2 1 Zo = z^ + — f 7t I b) z = a + bi => 71 ism-12 12 1-i COS 4a2 =0 a = 0->b = 0:z = ^ 12 a = —>b = - : z = - + - i 3 3 z = z => z la so' thuc Bai 10: a) Cho z = — + — i Hay tim 2 + a + (2ab-b)i = fa'-b'+a = ^ » < =>b = v b = 2a 2ab - b = J + ^ Z, Z, ^ Z, tiL=r b) Viet dang dai so so phuc: z = 71^ • 41 4i: — + — la mpt so thuc Z.Z 571 = cos — + ism — 6 71 isin — = COS 12 12 57t + —1 11: Giai cac phuong trinh tap hop cac so phuc y=x+5 / ,x 12 23 12 23 X = ( y - ) ( y - l ) ^ x = - — ; y - — ; z = -—+ — i Bai 9: Cho hai so' phuc zi va Z2 deu c6 modun bang Chung minh z - 1+ , =1 x(2y - 3) ^ ^- cos + 12 = 0y = x + Ket hgp ta c6: + yi, ta c6: o x - y H b)Tac6: i - v/s =-N/S + i = Giai = X + v/3 Vav zii = • ^ Bai 8: Tim so phiic z thoa man hai dieu ki^n sau: a) Dat z 3N/3 7? = \ I z I = va^ + b^ Phuong trinh tro thanh: Va^ + b^ - (2a + 2bi) + + Si = n«» A ^ * ^ riufi^ •Ja^+b^ = 2a - d) -2b + = ^ b = 4,a = hay a = - « Vay nghiem phuang trinh la: z = + 4i; z = — + 4i c) X2 -1 =5 =3^X = X f:V = ±i = ! |z + i | = | z - i x2 + (y + 1)2 = x2 + (y - 3)2 c:> y = ' ' ' d) Dat z = a + bi Phuong trinh tuong duang (a + bi)(a - bi) - 3[2a + 2bi + a - bi] =-19 + 3i x2 + y2 - 9x - y i = -19 + i X x^ +y^ - x = -19 z3 + = -9 + 9i •» (z + w)^ - 3(z + w) zw = -9(1 + i) Hay z + w = 3(1 + i) =^ zw = 3(l + i ) ^ + ( l - i ) ^^-7^ a + bi + 2a - 2bi => -24 + lOi 3a - bi = -24 + lOi => a = 8, b = -10 z=2+i w = l + 2i z + 2 = 26 d) z+i z-3i z = a - bi Phuang trinh tra thanh: = + Do do: z va w la nghiem cua phuang trinh: y =- l Vay nghiem la: z = - i ; z = - i a) i - z z + w = 3(1 + i) Bai 13: Tim hai so phuc z va w thoa man: ;; ±N/3 y = - ; x = hay ^ Tap hgp cac diem bieu dien la duang t h i n g y = : x ' ' - x - = o x = - l hay x2 = X^ = z+i z-3i - Giai a) z = - 3i + (1 - i)^ = - 3i - - 2i = - 5i = ^f29 b) z = - V3 i = s = cos + ism 2 Bai 16: Giai phuang trinh: z^ + (1 - 2i)z2 + (1 - i)z - 2i = biet rkng phuang trin CO mot nghiem thuan ao am nanj^ Iuy4n I hi OH - Nguyen ham - Tich phdn - So phirc r 2012 ^ ( 2012 ^ (2V2)2012 COS 71 + i s i n V I , Giai D|t z = bi la nghi^m thuan ao ta c6: Do do: z = b'i' + (1 - 2i)bV + (1 - i)bi - 2i = -b-^i - b + 2W + bi + b - 2i = b - b + (-b^ + b + b - 2)i = b - b' =0 b = l=>z = i -b'+2b2+b-2 =0 Phuong trinh tro thanh: (z - i)(z2 + (1 - i)z + 2) = z = i va z^ + (1 - i)z + = a)BaiViet 17: dang luong giac ciia so'phuc: z = Vs + i cos 7t b) Vs / 73-i =2 cos cos- Do do: z = c) z= 344 t (r 37C" cos— + isin — 6 571 571 ^ -71 -7t COS- + isin cos1 cos ; r + i sin 2Tt] , ; l + i = cos—+ Sin — 6) 37: cos + isin = 12 J V^ = 2^^f cos f V J, v = 8-8>/3i 7t '3j I I +6,i s i n 71I 6j= -2 COS V I 2; I , / / { - N , ,3018 N/3 + isin r COS 7t \( 7t -7 + isin — cos V 4j + V isin I 4j / ^ 371^ K z= -2 cos 71 7t = cos 28 + l , + isin 17 j iil9 7t 71 7t + isin f 2n' 7: z= -2 ^ cos I ' j + isin ^ 371 n cos n3- + ism 2-2' a)z: / 7t 7t Giai 7t 7t Ap dyng khai trien: (1 + i)"= |N/2J = cosn —+ isinn — 4j 57t N2011 Giai s + isin — " Sin — isin — 7t- / jilS: Tim phan thuc, phan ao cua so phuc z = + (1 + i) + (1 + i)^ + + (1 + \)^ ^ : 7:^ 7t 7t Viet dang dai so cua so phuc z = -2 cos—+ isin — c o s4— isin — V {V2-V6ir^ -,3018 l-i>/3 b) Viet dang dai so'cua so'phuc: z = (V3-i)^ c) Viet dang dai so ciia so phuc: z = - Iran r 1471^ + isin I 6; 1471^ V )) isin 37t 28 J J Cho so phuc z thoa man: (1 - 2i)z - ^—!- = (3 - i)z Tim toa d? diem bieu 1+i dien cua z mat phang toa dp (xOy) + Z2 zi Goi zi, Z2 la nghi^m cua phuong trinh: z^ - 2z + + 2i = Tinh Giai Phuong trinh da cho tuong duong: ({1 - 2i) - ( = i))z = i l i « (-2 - i)z = ^ 1+1 * ciia z la M o z = i Vay diem bieu dien JO'lO, 10 10 Phuong trinh da cho tuong duong voi: (z - 1)2 - (x - i)2 = (z - i)(z - + i) = 0z = ihayz = - i Do do: Z I + Z2 = + | - i | = N/S 345 20 Bai a) Cho so p h u c thoa man: (2 + i)z + b) Ta c6: + i N/S = 2(cos — + isin —) => (1 + i VB y'' = 8(cos7t + isinn) 3 ^^'^ = + 8i z = T i m m o d u n ciia so'phuc: w = z + + i ~ fZ, 8(cos7t + i s i n i ) —^ „ /T, 2v2(cos b) Giai p h u o n g t r i n h : i ? + 3(1 + i)z + 5i tren tap h o p so p h u c 371 +isin Dat z = a + b i , ta c6: z - (2 + 3i) z = - 9i a + bi - (2 + 3i)(a - bi) = - 9i - a - 3b + (3b - 3a)i = - 9i o , * Bai 21 Goi z,, z , la n g h i e m cua p h u o n g t r i n h : +zl+ Z3 H a y t i n h S= i z - = H a y vie't Giai dang l u g n g giac ciia z,, Z2 z''+ = o (z + l ) ( z - z + 1) = Suy Giai: i ; Z2 = - + I N / i , , „, 27t 27t z, = ( c o s - + i s m - ) va cua z , la: Z - , - ( c o s — + ism — ) ' 3 ' 3 , S = -] 2 57t 571 , +2COS 571 , , ; 57t , ) + isin( ) ' ^ = -1+1=0 Giai (3a - b - 2) + (a - 7b + 6)i = A A M M ' la t a m giac deu A M = A M ' = M M ' ^ ^ , a = l , b = l =>Z = + Z - = I Z2 - I = Z - Z M a t khac: I z^ - | = | (z - l ) ( z + 1) = I z - I w I = I + 3i I = T va I z^ - z I = I z(z - ) |z+1 = |z| | z - l | v i z 7t => z - ^ d o d o A A M M ' deu = z + 23 = Iz z I = o M tren d u o n g t r o n tam O ban k i n h R = a) T i m so p h u c z thoa m a n z^ = I z h + z |z + l | = z| (x + l ) + y = x + y2x = -— ( v o i z = x + y i ) b) T i m m o d u n ciia so p h u c z thoa man: (2z - ) ( + i) + ( z + ) ( - i) = - 2i 346 ^ l u g t ciia: 1, z, z^ Xac djnh cac diem M cho tam giac A M M ' la tam giac deu a-7b+6=0 Bai [ Bai 27: Cho z la so p h u c khac 1, goi A, M , M ' Ian l u o t la cac d i e m bieu dien Ian Datz = a + b i ( z ^ - l ) w =•! + z + z = + 3i => = - + cos — + isin — + cos( 3 Bai 22 Cho so phuc z thoa man ^^^^'^ = - i T i m m o d u n cua so phiic: w = + z + z +i 3a-b-2 =0 z, = S = (-1)'^ + ( c o s - + i s i n - + ( c o s ( - - ) + i s i n ( - - ) ) ' ^ 3 3 , Giai z, = - , z, = — + — 1; D a n g l u o n g giac cua z^ la: Taco: ^^^^^ = - i » z +i *, Bai 26: Goi cac n g h i e m ciia p h u o n g t r i n h z^ + = la z,, Z j , Z3 7t -a - 3b = 3b-3a = -9 a = , b = - l o z = - i ^^-(3.i)-(l-i)_^_ ^ ) b) A = - i = (1 - i)^ d o d o cac nghiem ciia p h u o n g t r i n h la: 7t ^ Giai z = + 2i=:i>w = + i = > | w | = N/4^+3^ = Ta CO A = d o d o z, = + 7t, 3:1, a) P h u o n g t r i n h t u o n g d u o n g : (2 + i)z = + 7i =-l-2i Bai 25 T i m so p h i i c z biet z - (2 + 2i) z = - 9i Giai z =l ( ^ i ^ ^ K — = N/2 (cos - + isin - ) = + i ^348 IT Cam nang Ivy^n thi DH - Nguyen ham - Tit h pluhi - So phi'rc - Trdn Bd Ha Cam nang luyCn ihi DH - Nguyen ham - Tich phan - So phi'rc - Tmyy Bd Vay cac d i e m can t i m la giao d i e m cua d u a n g th3ng Giai a) Dat z = a + b i , ta c6: z^ = a^-b^ =a^ + b^+a { 2ab = - b x2 + y2 = l ^ M i ( - - , + — ) , M ( - - , - — ) y ' 2 2 a = -2b^ Bai 28: V a i m o i so' p h u c z ^ dat w = — — , z+1 b(2a +1) = - u - n 1 Vay CO ba so phuc: z = 0, z = - \i "2* a + b - = -2 i(x + yi)^ — = x + l+yi a)z = x = y i = > w = b) Dat z = a + b i , ta c6: ( z - i ) ( + i ) + ( z +1)(1 - i ) = - i » a - b + (a + b - ) i = - i 3a - 3b = v o i z = x + iy b) T i m tap h o p cac d i e m bieu d i e n ciia so phiic z cho w la so'thuan ao 1 •^2''^^~2 , ? a) H a y viet d a n g dai so'ciia w a = 0, b = hay a = - - , b = - h a y a = - - , b = - y 2 ^ 2 ' U ~ ^ voi duong tron - b^ + 2abi I z h + z = a2 + b^ + a - b i , d o d o z2 = I z h + z W Ha w = a =— (x - y ^ ) i - x y ] [ x + l - y i ] ^ _ y x + x + y - ) ^ x^ + xy^ + x^ - y ^ (x + l ) - + y ^ b) w la so t h u a n ao nen b= -3 y D o d o I z I = yja^ + h^ = i(x'-y^+2xyi) — x + l+yi 0, X 7i - (x + l ) - + y = (X+ -y(x^ + x + y^) = : tap hop cac diem bieu dien la true Ox loai diem (-1,0) (x +1)^ + y ' = : tap hop cac diem bieu dien la duong tron (-1,0) ban kinh R = Bai 24 a) T i m so'phuc z thoa m a n : z - Bai 29: Cho z la so phuc khac 2i, dat f(z) = ^ ~ ^ z-2i - =^0 G p i M , A , B Ian l u p t la d i e m bieu dien ciia z; 1, 2i b) T i m phan t h u c phan ao ciia so phuc: z = a) Bieu dien d a n g dai so'va d a n g l u p n g giac ciia f(i) ^ l + i>/3l +i b) Giai p h u o n g t r i n h : f(z) = 2i c) T i m tap h o p (C) ciia cac d i e m M cho I f(z) I = Giai a ) D a t z = a + b i , ta c6: z - d) T i m tap h p p ( C ) cac d i e m M cho: acgumen ciia f(2) bang ^ ^"^'^-1 = Giai o a _ b i + ^±1^-1=0 oa2 + b2-5-iN/3-a-bi a + bi (a^ + b^ - a - 5) - (b + N/3 ) i = o o = [a) D a t z = x + y i f(i) a^+b^-a-5-0 i-1 i-1 - = — i-2i - i =i(i-l) =- l - I b + N/3=0 i a = - , b = - >/3 hoac a = 2, b = - Vs V a y z = - - Vi i h o l e z = - Vs i D a n g l u o n g giac: f(i) = 42 l b ) f ( z ) = 2i 347 z-1 z-2i = 2i 72 72 , z - = 2i(z - 2i) cos 37t + isin 3n Cly Z ( l ^ c) - 2i) = Ci> Z = _5(l+2i) l-2i TNHHAfTV D VVH Khang Viet r I 8^ —3 j Suy ra: Tap h g p cac d i e m M la du-ang t r o n tarn I ban k i n h R = V20 2V5 3 20 = - So phi'rc - Trdn ; Ha - i ; Z4 = - \/3 i ; zs = - - 2i H a y bieu d i e n cac d i e m M i , M2, M^, M va Ms Ian l u o t la d i e m bieu dien cua cac so phiVc tren, vie't cac so phuc ve dang l u o n g giac ^ ^ 2: Cho a la so thirc, xet so phuc: z = + cos2a + isin2a Tinh theo a, m o d u n va "3'3 acgumen cua z K h i a thay doi, t i m tap hop cac diem bieu dien cua so p h i i c z |i_3: Viet ve d a n g z = a + b i cac so p h u c b) Z2 = 1-i nen c) Z3 CO m o d u n la a) H a y bieu d i e n z va z' theo (z + z') va (z - z') va acgumen la l+ /, \ 2i (l + i ) ( l - i ) 371 „ , ^, , „ , , V3 i N/3 | i : Cho so phuc: z , = — + -> ^2= Y^~l ' ^/3 i V3 ^'""2~'T H a y vie't cac so p h u c ve dang l u o n g giac va t i n h tong: z^ + Z j + Z j + z^ va tich Z i Z j Z j Z ^ z- z b) G o i A , B, C Ian l u g t la d i e m bieu d i l n cua cac so p h u c z, z' va (z + z') hay bieu d i e n t i n h chat h i n h hoc ket qua cau a va c h u n g m i n h t i n h chat d o Giai C h u n g m i n h z^ = z^ = - va z^ = z^ = iL5: Cho da thuc: P = (x^ - x + 1)^ + (x^ - 2x + 3)2 H a y phan tich P t h a n h tich ciia cac tam thuc bac hai v o i he so'thuc , (z + z ' ) - K z - z ' ) , , (z + z ' ) - { z - z ' ) a) Ta co: z = ^ -^ — va z = — '2 '• , 1+ i a) zi = Bai 30: C h o z va z' la hai s o p h u c i 6: Cho a la so thuc, giai p h u o n g t r i n h : - 2(cosa + s i n a + isina)z2 + (1 + s i n a c o s a + i s i n a c o s a ) z - 2(1 + i)sina = 0, biet rang p h u o n g t r i n h nhan m o t n g h i e m dang a ( l + i) v o i a la so' thuc, k h i Theo bat d a n g thuc tarn giac (ve doan thSng), ta c6: a thay d o i t r o n g [0, 27t) T i m tap hop cac d i e m bieu dien ciia cac n g h i e m cua | z | < i ( | z + z'| + | z - z ' p h u o n g t r i n h tren Suy ra: < b) Theo gia thiet ta c6 I z + z' I = O C va Iz- z + z' + = OA; i_7: H a y phan t i c h tich cua hai thua so bac hai v o i he so thuc cac bieu I z' I = z' I = A B Bieu d i e n h i n h hoc cac vecto OB thuc sau day: A = x ' + x2 + b) B = x^ + 2x2cosa + c) C = x^ - 2x2cosa + i 8: a) C h u n g m i n h rang v o i m o i so thuc a, so p h u c z = ^ ^ ' ^ l u o n l u o n c6 ~ la O A , OB, O C la vecto b i e u d i e n cua z, z', z + z' d o do: O A C B la m o d u n bang h i n h b i n h hanh, ta c6: Dao lai, cho so p h u c z c6 m o d u n bang c h u n g m i n h rang c6 the bieu dien d a n g z = i i i i H a y t i n h theo a, acgumen cua so p h u c z 1-ia 350 Bd ,., 1: Cho cac so phuc: zi = + i ; Z2 = + i%/3; Z3 = Vs '•,1 i'iiiv T u gia thiel ta c6: M A M B Tap hop cac diem M la d u o n g tron d u o n g k i n h AB ,_, - Tich phdn • O A 4OB < +IA +IB = O C + AB BAI T A P L U Y E N T H I acgumen ciia f(z) = ( M B , M A ) z + z' ham t o n g cua hai d u o n g cheo cua c h i i n g V i z - la vecto bieu d i e n la M A va z - 2i c6 vecto bieu d i e n la M B < - Nguyen Vay: T r o n g m o t h i n h b i n h hanh, nCra chu v i l u o n l u o n n h o h o n hoac bang d) A c g u m e n cua f(z) = acgumen ciia (z - 1) - acgumen cua (z - 2i) Suy ra: thi DH OB < + IB Vay p h u o n g t r i n h f(z) = 2i c6 n g h i e m z = + 2i X - + yi - + y i = X + (y - ) i If(z)I = o x + (y-2)i 1^ o x+— + yJ K 3, luyen O A < O I + IA = l+2i (x - 1)2 + y2 = 4(x2 + (y - If) i 'c'lm nang i , 351 Cly TNHH MTVDVVHKhang Vi(>, Bai 9: Diing cong thi'rc Moivre de tinh cos5x va sinSx Ian lugt theo cosx va sirix luyen thi DH - Nyjp i n /hiin - Tichphdn - Sophi'ec - Tran Bd Ha iKing : a) Giai phuong trinh: (z^ + z)(z - 2i) - = z^ Cho z = {\[2 + \/6) + (\/2 tir suy gia trj cua s i n — va c o s — (Ichong dung may tinh) \hii - y[6)i Tinh z^ va suy dang lugng giac cua z 10 Bai 10: Tinh P = (cosa + isina)(cosb + isinb)(cosc + isinc) , HU'6NG D A N j T u suy cong thuc tinh cos(a + b + c) va sin(a + b + c) GIAI ' 1: z, = \/2(cos— +sin —) • ^ 4 Bai 11: Cho u = costp + isin(p va v = coscp - isincp 1) Tinh coscp, sincp theo u va v 2) Chon n e N ' , hay tinh cosncp, sinncp, theo u" va v" Z2 = ( c o s — + ism —) 3 3) Tinh c o s > , sin-^, coshp, s i n > theo cac ham so sin, cos ciia cac goc cp, 3(p, 5(p Z3 = N / ( c o s ( - - ) + i s i n ( - - ) Bai 12: Dat J = c o s — + ism — l)Tinhr, y J" ^4 571 / 57U^ = c o s — + isin — 2) Tinh + J , + J + J2; ^ J P 5n 57i' c o s — + isin — 4 3) Tinh theo n G N , tong so: + J" + ' 2: Ta c6 I z | = ^(l + cos2a)^ +sin^2a = + 2cos2a = V4cos^a = cosa 4) C h u n g minh dang thuc: (1 + a)" + (1 + Ja)" + (1 + y-af = i + c:\i' + c::a'+ + cV vi a thay doi t u - ^ den ^ thi ta c6 moi so phuc z, do ta c6 the viet: vai 3p la boi so cua nho hon hoac bang n 5) Suy gia trj S = + C^„„ + C^;,, + C;'^,, + + C;';;o = 2cosa voi - — < a < - vi + cos2a = 2cos2a va sin2a = 2sinacosa 2 Bai 13: Tinh b^ng hai each (1 + i)" va (1 - i)" suy Sn = - C ; + C^ - C;; + -f (-1)" C^-^ + Bai 14: C h o so phiVc z = x + yi, dat w = a) Tinh theo x, y modun i w de w Suy ra: z = 2cosa(cosa + isina) dieu c6 nghla la a la mot acgumen ciia z va gia trj S100 Goi I la diem bieu dien cua so thuc z-1 va M la diem bieu dien ciia so z +1 tu suy tap hop cac diem bieu dien M cua z bang so k cho truoc b) Cho I va J la diem bieu dien ciia cac s o + , - C h u n g minh I w MI tu MJ I M = va •2a ( I X , I M ) = 2a + k2n b) Cho phuong trinh: z" - 6z^ + Uz'- tarn I ban kinh R = ' 28z^ + 28z2 - 16z + = Chung minh rang cac diem bieu dien cua cac nghiem cua phuong trinh tren doi mot doi xung qua voi diem 1(1, 0) Giai phuong trinh va bieu dien nghiem (4,o u = sin — la nghiem cua p h u o n g t r i n h : Giai p h u o n g t r i n h (2) ta co nghiem: a = 0; a = cosa, a = 2sina Chi C O a = 2sina thoa man p h u o n g t r i n h (1), d o zi = 2(1 + i)sina la 16x^ - 20x-^ + 5x = c:> x l l 6x^ - 20x2 + 5] = nghiem, p h u o n g t r i n h t r o thanh: [z - 2(1 + i)sina] {z? - 2cosaz + 1] = Giai z^ - 2cosa + z + = Ta CO nghiem zi = (2) >ai 9: Theo cong thuc M o i v e ta co: (cos5x + isinSx) = (cosx + isinx)^ = cos^x + , D o n g nhat ca phan t h y c va phan ao ta co he: 2a a " - a ( c o s a + s i n a ) + s i n a c o s a 1+a^ acgumen cua z xac d j n h b o i : ^ = arctana + k2Tt + a(l - 4sinacosa + 4isinacosa) - 2sina] = -2a" c o s a + a(1 + s i n a c o s a ) - s i n a = 2a r^- , (t) 1-a^ a Dat a = tan—=> - = cos^, ;- = sm(j) 1+a l + a" + ia D o do: z = coscp + isin(p = 1-ia l-3i -• P = I ( x - 1)= +1 Khi C O S T : + isinn, z.i = cosTt - isinTt M l d i d o n g tren doan A B v o i A(2, 2), va B(-2, - ) dien M2, M la d u o n g t r o n tarn O ban k i n h R = la nghiem cua p h u o n g t r i n h : 16x^ - 20x^ = 5x = , t u o n g t u ta co: Goi M l la d i e m bieu dien ciia zi = 2(1 + i)sin7t, k h i n thay d o i tren [0, 27t] thi Goi M2, M l Ian l u g t la d i e m bieu dien cua Z2 va Z3 t h i tap h o p cac d i e m b\e^ cos5x = =:> v = cos^^ 71 7^0 + 275 71 „ ' f• 71 Vl0-2>y5 cos — = ; sin — > => sm — = 10 5 &ai 10: T h u c hien phep tinh ta co: *- ;' ' P = (cosacosbcosc - sinasinbcosc - sinbsinccosa - sincsinacosb) + 354 J, i(sinacosbcosc) + sinbcosccosa + sinccosacosb - sinasinbsmc) Cty TNHH MTV D VVH KhangVi^, So P la ti'ch cua ba s o p h u c c6 m o d u n bang va acgumen Ian l u g t a, b, c ne^ P CO m o d u n b a n g va acgumen la (a + b + c) D o d o P = cos(a + b + c) + isin(a + b + c) Cdm nang luyen thi DH - Nguyen ham - Tich phan - So phi'ic - Trdn Bd Bai 13: Ta co: + i = V2 (cos - + isin - ) = > ( ! + i ) " = (-Jl 4 Ha )"(cosn - + isinn - ) 4 (1 - i ) " = ( V2 )"(cosn ^ - isinn ^ ) D o n g nhat phan t h y c va phan ao, ta co: cos(a + b + c) la p h a n thuc va sin(a + b + c) la phan ao Bai 11: a) Giai he ta co: cos (p = ( l - i ) - l - C ; , i + C;i^-C^J^+ + (-l)"C^i" , sm{t) = , u " =cosn(j) + isin(j) b) Ta co: v " =cosn())-isinn(j) c) cos"* (j) = u"^ - v ' - u v ( u - v) 1^ - sin 3(1) + sin d) s m ' (j) = 2i , S , ,,5 (1) - — x + iy-1 x-yi-1 Bai 14: a) Ta co: w w = ^ ^ x + iy+ ' x - i y + D o do: N e u k;^ t h i (1) » cos 5^ + cos 3(j) + 10 cos (j) — d u o n g t r o n tarn I , • c sin5(t)-5sin3(t) + 10sin(i) T u o n g t u : sm^cp = 16 -'\~ b) Ta CO w = ] (lien h g p ) Sn = 1-k^ -2 1+ k 2^ 1-k^ X + = : Tap h g p cac d i e m M la ,R = k ,0 iz +1 I = MJ do MI I w I =MJ = k M I = K M J : tap h g p cac d i e m M la d u o n g t r o n (k 1) hay d u o n g Bai 15: a) Theo v i - e t cac so p h u c phai t i m la nghiem cua p h u o n g t i i n h : =0 (Z + - )2 = - 4i2 + (J2 + J + 1)"-? + D o d o cac n g h i e m la: s z+- = — 2 n = 3P + thi Sn = N/3 z+- = 2 b) Dat z' = + z ta co p h u o n g trinh tro thanh: 3) Neu n = 3k t h i J" = J^" = l ; l + Jn + j-S= + k^ v i P + J + nen: N e u n = 3P thi Sn = ; n = 3P + t h i S,, = + J = -f (1 + J'a)" = + c;j-a x^ + y2 I z - I = M I va Z2 + Z + + i s i n P i = - P = l +J + J2 = + J + P + + J" = (J2 + J + l ) n (1) th^nglj(khi k = l ) = d o do: N e u n = 3P t h i J" = 1; n = 3P + t h i J" = J; n = 3P + t h i J" = f 2) +J = c o s - - k^) + y ( l + k^) - x ( l + k^) + - k^ = 32 Bai 12: T) Ta co V = cosAn + isin47t = - ~ (x + l ) ^ + y ^ Neil k = t h i (1) ^ X = 0: Tap h g p cac d i e m M la true O y 2.,2, , ( x - l ) ^ + y^ = k ^ ( x - l ) + y2 = k2[(x + l ) + y2] w x2(l _ u^ + v^ + u v ( u ^ + v^) + 10u-'v^(u + v ) u +V V 8i^ (>/2)"oo6n^=l-Cf^+C;',- .+(-l)"CfP+ V o i n = 100, ta co Suw =-2-^" u-v^l^ cos^ (j) = cos , s i n n(t) = cos nij) = V i ( l + i ) " v a (1 - i ) " l a s o p h u c l i e n h g p n e n : u"-v" , V n> u"+ cos3(1) + c o s * , , T T (viu.v = l) ^ U V sin"* (j) = 2i u" +, v -5 + u v ( u + v) V ' + V ^ Theo khai trien N e w t o n ta co: (1 + i ) " = + C^^i + C ^ j ' + C^i^ + + Cj^i" , 72 + J) Giai (1): z'2 + = « z' = ±i N/2 ; z'4 + = z' = ± — (1 ± i) i[2'»"-1] Vay cac n g h i e m p h u o n g t r i n h l a : z = l ± i A / ; z = l ± — (1 + i) 35 Cty TNHHMTVDVVHKhanfi Bai 16: A p d u n g khai trien N e w t o n (1 + I ) " = C" + iC', + i^C^ + Viet MUC L U C + i"C" = C - •< - C ; -iC;^ c:, -iCf, = ( l - C ^ < , -C^; ).i(c^ -C-^ Phan 1: N g u y e n h a m - t i c h phan va l i n g d u n g Theo cong thuc M o i v e : (1 + i)" = yfl" (cosn - + isinn - ) 4 D o n g nhat phan thuc va phan ao v o i n = 2010, ta c6: "^2010 "-MIO + ' - K ) ^2010 + - - ^ +^2010 Bai 17: Ta co: ( l + iSJ" = 2^"c o s n - K h a i trien N e w t o n ta c6: U + isJsf" =(C -3CL ,2n -Sin +i(7ic^^ -37ic^„ + ^-3rC^-') cos I + isin I 1!! 3" 1- cos C" -ic^ +lct - v6iT= 2n D o n g nha't p h a n ao ta dirge: S = Vs v3y 2n sin- ^ ^2n " nn 'ai 19: a) P h u o n g t r i n h t u o n g d u o n g : z(z^ + l ) ( z - 2i) - (z^ + 1) = o (z2 Z^ + + isin 2n , / r , n V3 o + l)(z2 = - 2z.i - 1) = z2 + = hay z2 - 2zi - =0 z = ±i Z2 = i2 z - i z - l = c A = 0r:>z= — = i z2= 8>/3-8i = ] V a y z = cos - V6)2] + 2(72 + V6)(V2 - ^/6)i ^/i 1.1= 16 71 "12 + ism 16 Chuycn de3: Nguyen ham cua cac ham soca ban 26 60 Chui/cn del: Klidi niem tich phan 60 Chuycn del: Phuvng phdp tinh tich phan 87 C h u o n g 3: iTng d u n g cua tich phan 168 Chuycn difl: Dim tich hinh phdng 168 Chuycn del: U'ng dung tich phan delinh the tich cua vat the 189 cos Chuycn del: Tich phan lien ket + isin 210 110 Chuycn del: Phuang trinh hephuvng trinh tich phan 219 Chuycn dc3: Tich phan quy nap giai hqn cua tich phan 225 Chuycn de4: Bat ddng thuc tich phan 23S Chuycn de5: Tich phdn ham sohuu ti 251 Chuycn de6: Tich phan ham sovo ti 259 Chuycn de7: Tich phdn cac ham soluvnggidc 266 Chuycn de8: Tich phdn cUa ham so mil ham aologarit 273 Chuycn de9: Bin tap tong hap - bai tap luyen thi 281 Phan 2: S o p h u c Chuycn del: Khai nicni sophuc -phep tinh sophuc 30S Chui/cn de2: Can bdc hai cua sophuc phuang trinh bac hai 320 Chuycn de3: Dang hmiggidccua 327 sophucz Chuycn di}'4: Bai tap long hap ve sophuc Vay p h u o n g t r i n h c6 n g h i f m z = ±i ) Z = [(N/6 + SY-{42 Chiiyen del: Phuvng phdp thn nguycn ham C h u o n g 4: Cac v a n de lien quan phep tinh tich p h a n I = T=~ , C h u o n g 2: Tich phan 2n 3ai 18: Ta c6: - ^ ^2"*'^ =Y,^2ni'^^t k=() 2nn Chuycn del: Khai niem ngui/cn ham + isin2n — - C ; + + (-3)"Q^„") D o n g nhat ta c6: S = 2^" cos = C h u o n g 1: N g u y e n h a m = a + bi ; ^35 [...]... —) = - 1 2 Giai F(x) = sinx - cosx + C , 71 F ( - ) = - l ^ l + C = - l < : > C = -2 Vay F(x) = sinx - cosx - 2 50 x-1 Giai Giai F '(x) = f(x) Vx e R o (ax + b + - + C tim f(x) Giai = sinx + cosx F ( ^ ) = 0C = 0=s>F(x) = sinx-cosx 21nx Bai 42: Tim ho nguyen ham ciia ham so' f(x) = Tim nguyen ham F(x) biet F( —) = 0 f(x) = sin"* x 1 + cosx sinx dx Dat t = Inx => dt = X Cam nang luy$n thi DH - Nguyen... (-ax^ + {2a - b)x - c)e-" ham ciia f(x) = Vl + x" 48 49 Cam nang luyen thi DlI \'i;^iiyri; ham rich i>lh!ii So phin Iran Bd Ha Giai Giai 1+ F'(x)=^ Vl + x^ V l + x^ x + Vl + x^ Vl + x \ - V l + x^ = f(x), Vx 1 = G Do do: R f^=lnt +C J t dx = ln|lnx| +C xlnx Bai 41: Cho biet If(x)dx = In^x + C tim f(x): Vay F(x) la mot nguyen ham ciia f(x) Giai Bai 38: Cho f(x) = sin3x(l + cotx) + cos''x(l + tanx) (0... 0 lis giai thiet ta c6: 6 — 2 b+ c = 4 a + b = 0 Giai f(x) = cos^x - sin''x = (cos^x - sin-x)(cos2x + s i i i - \ f(x) = cos2x => F(x) = Ti \ 6 2 Giai h ^ ta c6: a = 1, b = - 1 , c = sin2x + C F(-) = 0 < » - s i n - + C = 0 « C 3 73 = ' Bai 5: T i m h a m so y = f(x) biet rling: Vay F(x) = — sin2x • ^ 2 Bai 3 Cho f(x) 5 4 V3 1 1 - f ' ( x ) = 4 N / x - X va f(4) = 0 4 1 V5x + 3 - V 5 x + l Giai... n ' x 0 3 Bai6: Tinh I = 2 3 a/2 JcosxVsin'xdx a/3 61 Cam nang luyen thi DH - Nguyen ham - Tich phdn - So phirc - Trdn Bd Ha Giai Giai 1= 7t/2 I cosx(sin x ) ' d x = a) A = J(x -1 + oVT^dx = - J(l - x)Vl - xdx + j V l - xdx (sin x)^ d(sin x) n/3 0 (sin x)^ I= 5 3 Tt/2 _3 /3 5 [ 2V4J = -(i-x)'vr^-^(i-x)Vr^ 4 Bai 7: Tinh I = dx.Tinh Jf(t)dt 3 Giai • +Ve 1 b) X I = |e-=^dx+je^dx = —!-je-'^d(-2x) + 2|e2d(-)... so'dang P(x) , i ; 3x- + 1 _ — 7 -2x- -5x + 6 Giai 3x' + l ^^^x- ~5\ 6 5.^ I T ? + Ne'u bac ciia P(x) nho han bac cua Q(x) th'i phan h'ch Q(x) thanh tich cac 3x' + l ' thuc don gian (mau so' la cac thiia so' bac nhat, bac hai 6 tren) de tim ho nguyen ham ~ T * „ 14 x = 3=>C= — 1 dx (x + a ) ' 1 l - k ' ( x + a )k-l ' '> +c + Chu y phirong phap dong nhat da thiic khi phan tich b) Bai 1: Tim hp nguyen... 27 C 7)' TNHH MTl nVVH Khang Viet Cam nang luy^n thi DH - Nguyen ham - Tich phdii - So pht'rc - Trdn Bd Ha b) Ta c6: +2x + 6 X x^-7x^+14x-8 +2x + 6 B (x-l)(x-2)(x-4) x-1 Giai C • + • x - 2 ' x - 4 Dat u = X + 1 => d u = dx - A ( x - 2 ) ( x - 4 ) + B ( x - l ) ( x - 4) + C ( x - l ) ( x - 2) dv = (x-l)(x-2)(x-4) dx (2x-l)' D o do: k = D u n g he so bat d j n h (gia trj dac biet) x = l r : > A = 3 ;... < —) |f(x)dx = In^x + C =^ F(x) = In^x ^ f(x) = F '(x) = Giai Giai 2 2 x-dx = — x ^ + C = - x V x + C y/xdx= + cos"^ x 1 + sinx cosxy Bai 43: Cho biet = sin^x + cos^x + sin^xcosx + cos^xsinx f(x)dx = 1 = sin^x(sinx + cosx) + cos2x(sinx + cosx) Vi F(x) = F(x) = sinx - cosx + C Bai 39: De F(x) = (ax + b)e'< la nguyen h i m cua f(x) = x.e" thi gia tri ciia a, b bang (x+ir Hay: f(x) = ' bao nhieu? = (x... + 6 x + 8=>x = 2(t + 3) V(2x + 1)' -V2X + 1 t^+6t + 8 , dt Giai dx B= J 2 (2x + l ) ' - ( 2 x + l)= 30 t-8 dx=l:±^d.,dod6E=i|-(*^^) 2(1 + 3)-^ t- t^-8 2(t + 3) 3i Cdm nang luy^n thi DH - Nguyen ham - Tich plidii - So phi'rc - Trdn Bd Ha E= f - ^ = lnlt + 3 + C = lni x + 3 + -\/x- +6\ 8 Jt + 3 ' (lyTNHHMTV phitt»ng phap a Dgng R(sinx, cosx) Giai dx= X= -^^-^ 2(t-2) 0-2)^+4 r-8 F= X = 1 - 2(t-2) (t-2)-^... Giai sin x - c o s x Bai 48: Tim nguyen ham ciia f(x) = sin X + cos x Giai f(x) = x>/x -1 =>F(x)= yx2>/x - x + C Bai 49; F(x) = In x-1 la mQt nguyen ham ciia ham so' nao? x+1 Giai F(x)= In x-1 F'(x) = (x+ir x-1 x+1 x'-l x +1 Giai Je"'" sin2xdx = |e''"'M(sin' x) = e'-"'" +C 1 Tim nguyen ham F(x) ciia f(x) thoa man man F(—) l-cos2x 6 Giai 52 X + cos x F(x) = -In I sinx + cosx I + C F ( - ) = -ln>/2 +C... (X +1) +C dx yJx + \+yjx-\ Giai V7 3 3 Vx+T-V^^ 2 dx = - (X + l)2 - ( X - 1 ) 2 + C Vx + 1 + V x - 1 2 3 1 54164: Cho ff(x)dx = - - I n coskx + C Tim f(x) k Giai dx Ox tai diem x = - xac djnh F(x) Giai F(x) = tanx + C D 6 t h i y = F(x) c ^ t O x t a i x = - o F ( - ) = 0 « C = - l 4 4 F(x) = t a n x - 1 54 ' Dat t = Vx + 1 x = t2 - 1 => dx = 2tdt 71 COS X -r , , " •' Giai 1 BaL59: Cho biet F(x) ... phap giai - bai tap ap dung - bai tap tu luyen Cuoi moi chvrong deu co phan Bai tap tong hop va Bai tap luyen thi bao gom cac bai tap nang cao duoc tuyen chpn qua cac de thi dai hoc va cac de thi. .. sinh cac ki thi hoc sinh gioi, ki thi dai hoc Rat mong s u gop y cua doc gia va dong nghiep de Ian xua't ban sau tot hon Tinh chat co ban: + Neu F(x) la mot nguyen ham cua f(x) tren D thi F(x) +... THUAW 17 Clyjmm Ccim nang liiyen thi DH - Nguyen ham - Tich phdn - So phiic - Trdn Bd Hd Giai a) Dat u = In I In(lnx) | MTV D VVHKhang VH 2^ 3^ du = i l l l i l E ^ j x In(lnx) Giai ^3^ du = x l