Phương pháp tính tích phân và số phức phần 3

Gidi > Phirang t r i n h hoanh dp giao d i e m : 2" = - X x = l ^ \ V T = 2" la h a m t a n g : Do X = la n g h i e m n h a t S = fir (3-x)-2'' 3x- dx 2" y VP = - X la h a m g i a m Vay / y = 2' Inx ln2 (dvdt) 53o| T i n h d i ^ n t i c h h i n h phfing gi6i h a n bdi cac diTdng y=|x^-4x + 3|, y = x + DH kiwi A/2002 Gidi Phirong t r i n h hoanh giao d i e m cua h a i diTorng x-4x S = 5r L + 3| = x + X = X = (x + 3) - (x'' - 4x + 3) d x - _x^^5x2^ _iL + x - x (-x^ + 4x - 3)dx 109 (dvdt) 267 531 T i n h d i e n t i c h h i n h t h a n g cong gidi h a n b d i (C) : y = xln^x, true hoanh va h a i dudng t h i n g x = 1, x = e DH Xdy dung -1997 Gidi Vi j^^lxln^ X u = hi^ X =:> dv = x.dx Uj = In xln^x > 0, dx = du = do I n ^ x.dx X 21nx , dx S= V = X => d u j = —In^x Ji X I n xdx dx X Lai dat dvj = x.dx Do : = S = •e • hi X 2 e V -.dx 2 e x 2• + -l)(dvdt) = -(6^ I532I T i n h dien t i c h m i e n gidi h a n bdi y = x, y = x + sin^x va hai dudng thSng CO Ta 0, X = X = 7t Gidi : S = X - (x + s i n x) dx = sin^ xdx f" (1 - c o s x ) d x = 2J X - sin 2x = - (dvdt) T i n h d i e n t i c h cQa h i n h phSng gidi h a n b d i cac dudng : I x^ y = J4 - — va y = 4V2 DH khd'i B - 2002 268 Gidi y > * * 1/ — y = 14 - 2 PhifOng t r i n h hoanh giao d i e m : - 4V2 x" - 8x - = x^ = 00 x^ = - (loai) X = ±2>/2 N h a n xet S = Vx e 4>/2 j-2%/2 4- x2 x2 4V2 -2V2; 2V2 dx = - —.dx - r2^ x2 4V2 dx dx D a t X = 4sint D o i can Vay I.= h = Vay dx = 4costdt X = 2V2 X = ^ * 4(1 + cos 2t)dt = + - sin 2t cos^ t d t = r2j2 t = x2 4V2 2J2 :dx = S = 2(Ii-l2) = I2V2 271 + - 3; = 71 + (dvdt) |534| Cho h i n h p h i n g (D) g i d i h a n bdi cac difdng y = —^—1 + X va y = — a) T i n h dien t i c h h i n h (D) b ) T i n h the tich vat thi tron xoay k h i quay h i n h (D) xung quanh true Ox DH Nong nghiep Hd Noi - 1999 269 PhiTOng t r i n h hoanh : a) Vay b) SD- pi r -1 ^ dx l l + x^; Gidi 1^ ^7t l + x^ 3, O X = ±1 (dvdt) Ca hai dirctng cong deu n^m tren Ox nen ta c6 : VD/OX = 35 71 f1f -1 V + x' ^ dx - T: -1 y -dx = V ^271 (dvtt) Tinh dien tich hinh phang (D) gidi han bdi cac ducJng X = 1, X = e, y = 0, y = In X 2V^ ' DH Kien true Ha Noi - 1999 Gidi Ta CO : 8,0, In X e = 2A/^ • e Vxds , dx = Vx In X X • = V^-2 f ' - ^ 2V^ ISSGI Tinh dien tich hinh phSng = V e - V ^ ' = (2-Ve)(dvdt) gidi han bdi y = (x + l)'^ va y = e\ = ( D ) DH Hue - 1998 Gidi Phifdng t r i n h hoanh giao diem (x + if xet f(x) = (x + / - e" C O : = e" c6 nghiem x = va ta f (x) = ( x + ) ' - e" > - e > Vx G (0; 1) !=> f dong bien tren (0; 1) => Vay f(x) > f(0) = (x + l ) ^ d x - S(D) = ^ + i f > e" (X f69 eMx = - e (dvdt) Tinh dien tich hinh phing gidi han bdi true hoanh, y = x^ - 2x va dudng thang x = - , x = DH Thuang mai - 1999 270 Gidi Ta CO : 2x = - Vay X = X = —CO X - x^ - 2x s X V f - X - + J X X +00 - (2x-x^)dx A = -(dvdt) 3 (x^ - 2x)dx + -1 f + (x^ - 2x)dx = = ,2 38| T i n h dien t i c h h i n h p h i n g gi6i h a n bdi y = x** + 8x 3 HV Bau chinh Viin vay = ~ X X — thong - 1997 Gidi Phuong t r i n h hoanh giao d i e m ciia h a i dudng 7-x _ x-3 " X- 8x Y~ x = X = x = Theo h i n h ve, ta c6 : S = V ^ V X- 8x 3 8x -1 + 4 3'"^T~3~x-3 x^ 4x^ 3 iL + 2iL _ X - ln(x - dx 3) = (9-41n4)(dvdt) 539I T i n h dien t i c h S gidi h a n bdi diidng y = sinx t r e n doan [0; K ] va true hoanh 271 Gidi J y = sinx Nhcf t h i t a c6 : Trirdc h e t t a t h a y di/dng cong y = sinx cat x'Ox t a i d i e m 0, X = 71, X = 271, X - 3n X = s = •371, sin x dx sin xdx - r27t s i n xdx + = (-cosx) " +(cosx) 2n va \ ,n.M '///" 1/ 37t X sin xdx 3ii -(cosx) 2n =6(dvdt) 401 T i n h d i e n t i c h h i n h phAng gidi h a n b d i cac y = sin IXI diTdng : y = IxI - 7t DHMaHd N6i - A/2000 Gidi sin x CO Ta CO Ta : : y = sin I x y = |x I neu x > I = s i n ( - x ) = - sin X neu x - TI = neu X - X - 7t neu x > X - 71 < dx = Rcostdt Doi can -R X= R x = t = il 'g(x)=-N/R^-X^ t = -^ 274 s = = 2R2 f2 V l - s i n H c o s t d t = 2R2 f cos^ = R2 tdt ~2 sin t ^ (l + cos2t)dt = R2 = 7iR2(dvdt) 2 546| T i n h dien t i c h cua h i n h elip (E) : — + = „2 a b Gidi Taco: (E) : ^ a + ^ b = b - a \ a > b y = + -Va^ a -b Ta xem (E) la hop cua h a i dudng cong : f(x)= ^ V a ^ - x ^ ; a g(x) a Suy dien t i c h ciia h i n h elip E a dx = J-a L D a t X = a sint => dx = acostdt Vay S= b fa 2Va^-x^dx a J-a X = a D o i can X = -a t = ^ t = -^ — | \ V a ^ ( l - s i n H ) a cos t dt = 2ab J cos^ t dt S = 54?! 2ab x ( l + cos2t)dt I A t + - sin 2t = ab 7t = Trab (dvdt) Goi S la dien t i c h h i n h ph&ng gidri h a n boti y = ax^ va y = — ax^, hai ducfng t h i n g y = 1, y = (vdi x > 0) 275 a) Tinh S a = b) Tinh tat ca cac gia t r i ciia a (a > 1) cho S dat gia t r i Idn nhat Tinh gia t r i Idn nhat DH Hang hdi - 1998 Gidi a) K h i X > 0, a > t h i y = ax y = -ax dy = S = Va va V - I f2 2(V2^) Vydy = 3VI VI S = ^(5-3A^) 3VI Khi a = thi S = — (5 - 3V2) (dvdt) b) S = - ^ ( - V ) 3VI Dodo - S„,a, o Luc S„ax = a^i„ - o a=l (5 - 3V2) (dvdt) |548| Tinh dien tich hinh p h i n g gidfi han bdi : 2y = x^ + x - va 2y = -x^ + 3x + DH Hang hdi - 1997 Gidi PhifOng t r i n h hoanh giao diem ciia hai dUcJng cong : x^ + x - = - x ^ + 3x + c:> x^-x-6 =0 S = - ( x ^ + x-6) (-x^ Ta c6 : X x^ - X - x= -2vx=3 +3x + 6)dx = -2 -00 (•3 x^ - -2 X - dx -foo WwM - 276 l l l | Cho h a i so' phiJc khac l a z = rCcoscp + isincp) va z' = r'(cosip' + isincp) (r, r', cp, if)' M) T i m dieu k i ^ n can dii ve r, r', cp, cp' de z = z' Gidi Ta CO z = z' k h i va chi k h i : + Hoac r' = r, (p' = cp + k2n + HoSc r' = - r , cp' = + (2k + DTI (k e Z) (k e Z) 112i Xac d i n h tap hop cac d i e m t r o n g m a t phSng phufc bieu d i l n cac so phufc z t h o a m a n cac dieu k i ^ n sau : a) M o t acgumen cua z - (1 + 2i) b^ng - b) M o t acgumen cua z + i bang m o t acgumen ciia z - Gidi a) Goi z = X + y i t h i z - (1 + 2i) = (x - 1) + (y - 2)i N e n tap hop cac diem t r o n g m a t p h a n g bieu d i l n so' phufc z thoa yeu cau b a i toan la t i a c6 goc A (A la diem bieu dien + 2i) v d i vecto chi hudng u bieu dien so phiirc Vs + i (tufc la u c6 m o t acgumen la -) k h o n g lay d i e m A ( H i n h 1) O V3 Hinh b) Goi z = x + y i Hinh => z + i = x + (y+l)i => z - l = x - l+yi X e t d i e m B ( l ; 0) bi§u d i l n so 1, d i e m J(0; - ) bieu d i l n so - i t h i t a p hop can t i m 1^ cac d i e m thuQC dudng t h i n g B J n ^ m ngoai doan B J ( H i n h 2) 385 l i s j Tim p h a n thiTc v a p h a n a o c u a m o i s o phufc s a u : a) c) COS + if (V3 I isin— i^{l + V3i)^ b ) a-i)HS + if 3^ d)-"°°->^bietrkngz.l=l z ^ Gidi , m 7t a ) Ta CO : 7t Vs c o s — ism— = 3 cos 1; i = i + Si) = [1 + 2Si + 3i^fa + Si) (1 + Vsi)^ = [(1 + Siffa Vay - 721^ + 24V3i - 8)(1 + Si) n = [2Si n ' 2fa + Si) =S(24V3i^ i^(l +Vsi)"^ = — + —1 (64 + 64Si) = + 1281 2 isin— 3 V a y phan thifc bkng 0, phan ao b^ng 128 b) z = ( l - i ) ^ ( V + i)^ Ta CO : - i = >^ s s: 71 71 c o s — ism— A) (1 - i)"* = (V2)^cos7: - isinTt) = - [S T a CO : (>/3 + i) = 12 + —1 = cos— I 2; + isin—6J (1 - i)^ (V3 + i)^ = (-4)(-64) = 256 Do : (VS + i)^ = 2*^(cos7T + isin;:) = - ^ Vay s o phufc (1 - i)'^{S + i)^ c6 p h a n thiTc b k n g 256, p h d n a o b a n g {S c ) T a CO : (1 + i) = V2 — + iS = s[cos— + isin— 71 , ^ (1 + = 57t (>/3 + i) = 2 71 S"' c o s — + i s i n j V 'S T a CO : — 57t 4; = 321 1.1 = cos— + i s i n - +- 971 971 cos— + isin— (V3 + 1)^ = 2^ 6 = 2^ 371 c o s — + isin 371 2) = -5121 386 10 32i (1 + i) Nen (V3 + i ) ^ -512i Vay p h a n thuc l a d)Tac6: 16 z + - = l z 16' , p h a n ao l a z^-z + l = A = - = -3 = 3i' z = + V3i z = = — + — 3 2000 2000 + isin— COS— z^""" = cos z 0 071 271 + isin 271 cos 1 ,2000 271 271 cos— + isin — z2°*^° + 271 + 333,271 271 isin— 3 27C 2Tt 271 — — = cos— + i s i n — + cos Do so phvjfc z T i n h (1 - 20 0 - = — 3 z^ooo 114 20 0071 vi = C O S — + isin — 3 Vay 1-V3i z = A/3 z = +Vsi 3 + 27r 271 i s i n — = 2cos— = - 3 c6 p h a n thirc bang - va p h a n ao b ^ n g iSf Gidi Ta CO : Vay - iVs = cos - —— + i s i n v 3v f (1 - iVs)^ = 2^ cos - 71 ' - — + isin 3y 71 ^ V —3 , - 7t ' —3,; = 2'^[cos(-27:) + isin(-27t)l = 2*^ = 64 (cong thiJc Moa-vrcf) 115 27r 271^ Cho z = cos— + i s m — T i m cac so phiic (3 cho P'' = z 3 387 lisl Gidi D a t P = r(cos9 + isincp) t h i p ^ = r^(cos3(p + isinScp) Vay p^= z 2K 2n r^(cos3(p + isinScp) = c o s — + i s i n — 3J r = 271 , 3cp = — + K.2n, k e Cho k = 0, 1, 2, t a diTcJc cAc gia t r i khac cua P 1^ : Pi ^ 2n 2n^ cos— + ism9 J V 871 BTI^ cos— + i s i n — P3 - 3/ cos 1471 j 147:^ + isin 9 T i n h t i c h a p v a thucfng ^ v d i a) a = V2 cos— + i s i n — 3) 3n b) a = VS c o s — + i s i n - P= 3n'] , V Vs 71 n cos— + i s i n — 4) 471 P= c o s — + i s i n , 47r 3 ; Gidi a) T a C O : a p = V2.V3 cos CO : b) T a CO : Ta a - P = a.p = V2^ V3 cos ^ TT 2>/5 cos 4J 7t > 71 ^ 71 cos V 3, \2^ 47i' 3n 471^ ' K 37t 71 + 12 771' 77r cos— + i s i n — 12 12 j 47C 3)) 571 571 = 2V5 C O S — + i s m — j + isin 12 cos— + i s i n — -I ^7t_7I^^ + isin 1771 + isin f3n — H + isin + isin 1771 = 2V5 C O S p " f3_n _ V Vs 4K\\ 71 Til cos- + i s i n 6) " 388 l l ? ! H a y t i n h t i c h ap v d i : a ) a = coscpi - isincpi va P = cos(|)2 + isin{p2 b ) a = cos(pi + isincpi P = cos(p2 - isin 0, so da cho c6 dang lugng giac la : - 2sin K h i sin u -cos + — COS 4) + — 4; + isin — + — 12 4, < 0, so' da cho c6 dang lircrng giac : 12^4, - s i n q) VI n cos = cos 2^4 (p 57r + isin 2^Tj, -sin cp ST: 71 = sin 2^4, (p 57: ,2^T; 1191 T i m so phufc z cho | z | = | z - | va mot acgumen cua z - bang m g t acgumen ciia z + cong v d i — Gidi Goi z = X + y i (x, y € R) t h i |z| = | z - | c:> ^|x^ = - 2f + L l ^ , l ^ i y - , z l l i y ,-3^y^+4iy Khido z+ l + iy + + iy o x = l ^ ^ + y' {I la so' thuc ducfng) o y > Vay 120 y = V3 + Vsi z = Cho so phufc z CO modun b k n g B i e t m p t acgumen cua z la cp, hay t i m m o t acgumen ciia m i so phufc sau : a) b, - 2z2 d) -z^z c) ^ 2z f) z^ + z e) z + z h) z^ + z - z g) Gidi Theo gia t h i e t t a c6 z = cos(p + isincp a ) 2z^ = 2(cos(p + isincp)^ = 2(cos2(p + isin2(p) V a y 2z^ c6 m g t acgumen la 2(p 390 b) Ta C O : 2z = 2(cos(p - isincp) 2z - 1 (coscp + isincp) 2(coscp - isincp) (coscp - isincpXcoscp + isincp) — ^ = —[-coscp - isincp] = —[cos(cp + n) + isin(cp + n)] 2z 2 => Vay —\ CO m o t acgumen la cp + TT 2z c) z coscp - isincp - = z coscp + isincp (coscp - isincp) (coscp + isincp)(cos(p - isincp) = cos2cp - isin2cp = cos(-2cp) + isin(-2cp) Vay - CO m o t acgumen la -2cp z d) -z^.z = -[cos2cp + isin2cp][coscp - isincp] = -[cos2cpcoscp - icos2cpsincp + isin2cpcoscp - i^sin2cpsincp] = —[(cos2cpcoscp + sin2cpsincp) + i(sin2cpcoscp - cos2cpsincp)] = —[coscp + isincp] = cos(cp + T:) + isin(cp + TI) Vay -z^.z e) C O m o t acgumen l a cp + TI z = (coscp + isincp), z = (coscp - isincp) z + z = 2coscp • z.+ z c6 m g t acgumen b k n g neu p h a n thuc ciia z diTcfng C O m p t acgumen bSng TI neu p h a n thuc cua z a m C O acgumen khong xac d i n h neu z la so ao (tufc la z = i hoac z = - i ) f) Ta c6 : z^ = (cos2cp + isin2cp), z = coscp + isincp => + z = (cos2cp + coscp) + i(sin2cp + sincp) 3 2 3cp n acgumen cua z + z l a — neu cos— < z^ + z = -2cos— -cos Sep 3cp ism— 391 g) + = -2cos- acgumen cua T a c6 : COS — l2 + 7r J + isin I — + 71 ; + z k h o n g xac d i n h neu c o s - = (tufc 1^ z = - ) z^ = (cos2(p + isin2(p), z = coscp + isincp z - z = (cos2(p - cos(p) + i(sin2(p - sincp) ^ • 3^ • 3u) (p „.(!)'' 3(p 3(p' = - s u i — s i n — + i c o s — s i n — = 2sin— - s i n — + i c o s — 2 2 = s i n ^ cos I + isin ) f3q) + ^ 7i' , Do t a c6 : + ,, acgumen cua z - z la 3(0 + 71 (D neu sin— > 2 ,, + 3U) - TT acgumen cua z - z l a — (D neu sin— < 2 + acgumen cua z"^ - z k h o n g xac d i n h neu sin— = (tufc l a k h i z = 1) h ) T a CO : =:> Do t a z^ = cos2(p + isin2(p, z = coscp - isincp z^ + z = (cos2({) + coscp) + i(sin2(p - sincp) 3(p (P 3(0 u) „ 3(0 (p (p = 2cos—cos— + i c o s — s i n — = c o s — cos— + i s i n — 2 CO : acgumen cua z^ + z l a + 3(p acgumen cua z^ + z l a — neu c o s — > + *P \ 3(p „ — + 71 neu c o s — < j v2 + 3(p acgumen ciia z + z k h o n g xac d i n h neu c o s — = l | a) H o i v d i so n g u y e n dUOng n n a o , so phufc 3-V3i^ lV3-3i^ la so thirc, l a so ao " b ) Cung cau h o i tuong tiT cho so' phiic r 7+i 4-3ij 392 Gidi a) T a c6 : - V i _ (3 V3 - 3i nen SIKS + 3i) _ (V3 - 3i)(V3 + 3i) ~ b) >/3-3i n7i So' l a so' t h u c sin— = Soddlasoao cos— = n7i „ n = 6k + Ta CO : 12 S + i ~ 3-A/3I A/3 71 71 -= = — + - i = cos— + i s i n — N e n v d i so n n g u y e n duang t a c6 : • ^ + 6i _ + i - 3i 3i) ^ = (4 - 3i)(4 + 3i) (7 + i)(4 + N e n v d i so n n g u y e n dUdng t a c6 : 3-V3i n7t Vs-3i • S o l a so' t h u c S o Ik so ao HTI C5> n = k (k n g u y e n duong) o — = — nTi 71 , +k7i (k l a so' n g u y e n k h o n g a m ) f-( V2 + = 7t COS— + , =0 o 71 4J +i^ isin— nn nTi COS— + ism— 4 U-3i • = COS— + isin— 6 nTt o sin— o n = k (k n g u y e n duang) cos— = o n = k + (k l a so' n g u y e n k h o n g a m ) o — = = k7t - + k r I22I C h o A, B , C , D l a bon d i e m t r o n g m a t p h S n g phuTc theo thiJ tir b i e u d i e n cAc so + (3 + V3)i; Chiifng m i n h rkng + (3 + V3)i; + 3i; + i bo'n d i l m n a m t r e n m o t diTcfng t r o n Gidi Ta CO A(4; 3+ S); B(2; + VS); C(l; 3); D(3; 1) G o i ( C ) l a dudng t r o n t a m I ( a ; b), b a n k i n h R c6 phuong t r i n h : (X - af + (y - b)' = 393 (4) (3 - af + ( - b)^ = R ^ D e (C) • (3) ( - a ) ' + (3 - b f = R ' e (C) C • B e (C) • A e (C) o o » (4 - a)2 + (3 + V3 12 - a = L a y ( ) - ( ) t a CO : (2-af => T h e a = 3, b = v a o ( ) =o T h e a = v a o (1) v a ( ) + {3 + S « - b)' = R ' - hf = R^ (1) (2) a = b = R = Thur l a i a = 3, b = 3, R = t a c6 (4) d u n g V a y b o n d i e m A , B , C, D n a m t r e n ducfng t r b n t a r n 1(3; 3), b a n k i n h R = |l23| B i e u d i i n h i n h h o c cac so' + i v a + i , r o i c h i i n g m i n h r k n g n e u cac so' thiTc a, b t h o a m a n cac d i e u k i e n : < a < - , < b < - v a tana = - , tanb = t h i a - b = — 239 Gidi D i e m M b i e u d i i n so' + i D i e m N b i e u d i i n so + i M thi tan(Ox; OM) = tan(Ox; ON) = = tana o N 239 X = tanb 239 D o M , N n a m t r o n g goc p h a n t U thiir I c i i a h e t o a dp O x y Con < a < — , < b < — n e n m o t a c g u m e n cua + i l a a, m o t a c g u m e n cua 239 + i l a b TiS m o t a c g u m e n c i i a 3^9 + i l a a - b ( d a n g I t f p n g g i a c cua so phiJc) T a CO : nen 239+ i 239+ i 476 + 480i (5 + i ) * (5 + i)'^ 239+ i ma (239 + i ) ( l + i) = 238 + i = 2(1 + i) So ( + i ) CO m o t a c g u m e n b ^ n g - Vay 4a - b = - + k27i ( k G Z ) D i thay < b < a < - Suy r a 4a - b = - 394 124| Cho t a m giac deu O A B t r o n g m a t p h i n g phufc (O l a goc toa do) Chufng m i n h rkng neu A , B theo thuT tU bieu di§n cac so Z Q , Z i t h i ZQ + Zj = ZQZI Gidi • T a m giac O A B la t a m giac deu k h i va c h i k h i OA = OB va goc (OA,OB) bang - hoac - — tufc l a k h i va c h i k h i ZQ ^ va neu d a t — = a t h i 3 • zo i a I = va m o t acgumen cua a la — hoac -— 3 • M a t khac, k h i — = a t h i : ZQ + Zi = ZQZI O ZQ + a = o a - a ' + = a^Zg = aZfl ± V3i o + I = a I a = va m p t acgumen cua a l a — hoac - - 3 • V a y t a da chiJng m i n h O A B l a t a m giac deu k h i vk c h i k h i ZQ + Z j 125| a) Cho z - coscp + isincp - Z o Z i (zo # 0) (cp e R) Chijrng m i n h r ^ n g v d i m o i so' nguyen n > t a c6 : z" + — = 2cosn(p; z z" — = 2isinn(p z b) Txi cau a), chiing m i n h r k n g : cos'*(p = -(cos4(p + 4cos2(p + 3) sin^(p = ^(sinScp - 5sin3(p + lOsincp) Gidi a) T a CO : Nen : z" = cosncp + isinncp; — = cosncp - isinncp z z" + — = 2cosn(p; z Dac b i e t : z + - = 2cos(p; z z" — = 2isinn9 z z - - = 2isin(p z 395 Ta • Ta b)« CO z + - = 2cosq) z cos^cp — 1r 1^ — z + 4 cosq) = — z + - ^ ! z" + — + z ^ - + z ^ — + z^ 2^ + 4 z.— z^ + — [ c o s ( j ) + 4.2cos2(p + 6] = - ( c o s ( p + 4cos2(p + 3) 16 CO z - - = 2isin(p z If sine) = — z 2i 1^ — Z/ n5 • z sin^cp 1\ ' z V 2^i zj 1^ — z; s i n ^ ^ = - ^ ( s i n ( p - 2C5 sinScp + 2C5 sincp) 16 126i (sinScp - 5sin3(p + lOsincp) T i m d a n g lUcfng g i a c c u a cac c a n b a c h a i c i i a cac so' phufc s a u : a ) cos(p - isin(p b ) sincp + icoscp c ) sincp - icoscp v d i cp e R c h o t r U d c Gidi a) T a CO : coscp - isincp = cosC-cp) + i s i n ( - c p ) n e n d a n g lUcJng g i a c c u a c a n b a c hgii c i i a coscp - isincp l a cos b) T a CO : ^ cp^ + ism va sincp + icoscp = cos cos I - - + ism + 71 I - - + 7t , n + isin - - C P n e n d a n g l u c f n g g i a c c u a c a n b a c h a i c u a sincp + icoscp l a cos c) 7t cp 71 isin — u T a CO : va 2y sincp - icoscp - cos = cos cos - - - C P f57I u isin 2y + isin fn + isin " - ! f5n u ^-2 396 nen dang lUdng giac cua can bac hai cua sincp - icosip la cos + va isin cos f ^ 37r^ ; 2,n\ + ism "^T Viet cac so' phiic sau dudi dang luong giac : , -2 71 Tt cos isin — 4 cos— + ism— 6j 2I Ta b) 71^ 71 If c) a) 71 7t -cos— + isin— 3 a) Giai 2n + isin 2n -cos—3 + isin — = cos— — 3 7r CO : 7t Vi cos— = -cos—, sin— = sin— (hai gdc bu nhau) b) Ta CO : 3 71 -2 C O S — 7: = isin— ^ 71 -COS— + isin— A) 7r = ^ 71 71^ =— =— + i s i n6,- ^cos-6 - isin— 128| Tinh phan thUc va phan ao ciia so phiirc sau : c) Ta a) c) a) b) Ta CO : z= - - A/2 7t , 7: COS— z = V3 cos CO 71 ^ + isin— b) z= 371 COS \ v — + isin 6y 71 71 3 ( I 6, ^ cos— + i s i n - 371 isin— 2 : z = V2 f 37T + isin— / J7t> COS- 371 COS— 71 I COS— j Gidi 71^ + isin—4 , rv2 +1— 2j = 1+i Vay z CO phin thuc la 1, phdn ao la 1 iVs 71 = —+ Ta : z = cos-7r + isin— = + iV3 2 3; CO Vay z CO phan thUc la 1, phan ao la Vs c) Ta : z= 37t 371 COS isin— = V3(0 + i) = Vsi 2 Vay z CO phan thuc la 0, phan ao la V3 CO A/S 397 MVC LVC TICH PHAN ChUffng HO N G U Y E N HAM ChUcfng T I C H P H A N X A C D I N H 68 PHLfONG P H A P TICH P H A N T I T N G P H A N 52 P H U O N G P H A P D O I B I E N SO 39 K I E N THtfC CO BAN TICH P H A N C A C D A N G T H U C J N G GAP 231 Tich phan truy hoi 165 Tich phan cac ham lifgfng giac 155 Tich phan ham chiia gia t r i tuy^t doi 119 Tich phan cac ham v6 t i 101 Tich phan ham hufu t i Chitang D I E N T I C H H I N H P H A N G , T H E T I C H V A T T H E T R O N XOAY 283 B THE TICH VAT THE TRON XOAY 261 A CONG THLfC T I N H D I E N TICH H I N H P H A N G ChUimg B A T D A N G THLfC T R O N G T I C H P H A N 296 SO PHlTC 328 BAI TAP 326 K I E N THtfC CO B A N 398 y y y y X y y •vvw w n H a s a c; h h n g a n c o m V n Email: nhasachhongan@notmail.com 20C N g u y i n T h iMinh Khai - Q.1 - T P H C M BT: (08) 38246706 - 39107371 - 39107095 • F a x : 39107053 \ I' H O C T6T \ \ \ \ HOC TOT \ \ HOC TOT G I A I T O A N BAI GIANG i iiaiticii IU0N6GIAC !?i'l'"'!SS - N SMINH T A M , 245 Tran N g u y e n H a n- H P* D T : (0313) L U Y $ N T H I Di!hl HOC ^ ^ i [...]... ciia hai dudng : 2 , 4 = - V 4 - x^ — 9 =4-x2 o x*+9x2 -36 = 0 = ±V3 S = f ,.2 ; dx = 2 p /3 / ^0 2 3 dx 279 I554I S = 2 ( Tinh I = 2 73 V3 + 2 V4 - 73 3 3 cos^ t dt = 4 ^ S = 2V3 73 Jo dx D a t X = 2 s i n t => dx = 2costdt I = 8 ^ Do do D o i can : V3 X = x = 0 3 ( l + c o s 2 t ) d t = (4t + 2 s i n 2 t ) 47: + + 3V3 47: + = V3 3 3 t = 0 47I + 3V3 (dvdt) T r o n g m a t p h i n g Oxy, t i n h dien t... f '(x) = ^^^^^ ^ ^ j ^ ^ smx 6' 3 ^ Q ^ 71 71 6' 3 Ta x e t T(x) = xcosx - sinx T'(x) = cosx - xsinx - cosx = - x s i n x < 0 T(x) g i a m t r e n 71 Vx e 6' 3 n 6' 3 x 7t 71 6 3 - f' 3 f Vay 3V3 7t 3 s i n x 3A /3 - > > — 7t X 271 299 3 ^ dx < 27t n dx < 3 J^ X 271 Pha dau bkng : XQ = — t h i 4 •Tsinx 3 1 X 6 J aVs 1 , ^f3 Nen ta c6 : — < - dx s i n XQ 0 => ^ < r 4 3 o =^ 1 sin X x , 1 dx < 2 6 ^... 4 4 Suy ra 2 37 1 -dx< 2 fT dx ^ 3- 2sin2x f3n 2 [ 4 71^ 1 7t — < dx 4 4y < ' 4 7 dx 3- 2sin^x 4 Vay 3 - 2 sin rT ^ dx 37 t 71 T~ 4 71 3- 2sin2x"2' 4 297 5 8 l | Chiing m i n h r k n g : — < dx 0 5 + 3cos^ X 10 Gidi X e t h a m so f(x) = Ta CO : 5 + 3 cos^ — < 16 X 0 - < 5 + 3 cos^ x dx 0 5 + 3cos^x 0 < 3cos x < 3 5 dx _ ^ 4 P d x + 3cos^x 5 Jo 16... 278 Ta CO : S = x +2 dx -2 4 3 -2 = -(dvdt) 3 I552I T i n h dien tich hinh p h i n g gidi han bdi hai dudng cong : = ax va = ay (a > 0) Giai T a CO hai dudng cong (P) c i t nhau tai (0; 0) va A (a; a) 0 y = ^[s^ ,2 Vax dx a r- I - — Vx a v x 3 3 3 y 3 A 3a 3 553I T i n h di^n tich hinh p h i n g gidi han bdi cac dudng : y = - V 4 - x^ va x^ + 3y = 0 BH Bach khoa Hd Noi -2001... giao diem cua (Ci) va ( C 2 ) la : A(4; 2A/2), B ( 4 ; -2^2) Ta CO : 27y' = 8(x - if o y = ± 2A/2 3V3 (x - 1).A/X - 1 vdi X > 1 Do true hoanh la true do'i xilfng cua (Ci) va ( C 2 ) nen : S V2xdx+ r =2 A/2I-^V(X-1)' ix J = 2 0 V2^dx - ^ r Ji V3 ^|{x-lfdx 4V2 5 32 A/2 8 V2 „ /- 3 2 V 2 _^.1£9V3 = 3 15 Vs 3 72 /88 V2 = — V 2 ( d v d t ) 15 15 5561 Cho diem A tuy y tren (P) : y = px^ (vdi p > 0) Goi (D)... kien x > 0) lnx = 0 o (xlnx) dx = n => du = -2, In^x x = l x^ In^ xdx 21nxdx dv = x^dx => V = - - x^ In xdx 3 Ji x^Inxdx dx => du = — , Dat u = Inx dv = x^dx => v = X x — 3 - - x^dx = — I n x 3 Ji x^ Inxdx = — I n x I = Nen Tt(5e3 - 2) V = Tc ^ - A ( 2 e 3 1 ) 3 27 27 -ix3 ' 26^+1 9 (dvtt) |5 73| Cho D la mien gidi han bdi 4 ducrng y = 0, y = Vcos'' x + sin'* x , x = —, x = TI 2 Tinh the tich ciia khoi... ( x 2 - X i ) - - ( x ^ - x ? ) = - 6 - X i ) 3 k (x2 + X i ) + 6 (y - k x g ) - 2 (xg + X i + X j X g ) I- 4^ 3k2 + 6 ( y o - k x o ) - 2 ( S 2 - P ) = J V k ' - 4 k x o + 4 y o (k^ - 4kxo + 4yo) b = i(k2-4kxo+4yo)2 D 3 = ^ [ ( k - 2xo f + 4yo - 4x21i > 1 ^^^^ _ ^,^2 b 3 b Dau " = " xay r a k - 2xo = 0 3 Khido S„,„= i 8 ( y o - x 2 ) 2 b k = 2xo 3 =l(yo-x2)2 6 Cho (P) : y^ = 2x va di/dng t... 2"' < 2 2 3 2" 1 3 2" dx < -1 f1 2 " ' < 2" '"^^2'''dx< | _I \ 2 2.dx c -12 Vay 1 < x'^ < ^^l.dx => l a h a m so t a n g t r e n R , do do 2x -1 dx < sin x < 1 - < sin^x < 1 2 N h a n b a t d i n g thuTc t r e n cho ( - 2 ) , t a c6 - 2 < -2sin^x < - 1 Vi vay 1 < 3 - 2sin^x < 2 3it D o do... Si M a t khac t a c6 : ( 2 p a x + b - px^ )dx = pax^ + bx - px pa(xN^ - XM^) + b(xN - XM) - ^ (XN^ - XM^) 3 : (XN - X M ) :VS2 -4P S2 a 2 pa(xM + X N ) + b - | ( x ^ + x ^ i paS + b - ^ ( S 2 2A " (2 —a p+- b + - 3 3 j +XMXN) -P) 3 2a2p + b - 2 V ^ = 2 Do do 4a + — =1 + ^(pa^ + b) 3 \ =-Si 3 282 B THE TfCH VAT THE TRON XOAY KIEN THLfC C d B A N 1 Cho h i n h p h i n g gidi h a n b d i cac dudng... chiif n h a t A B C D quay quanh Ox V l a the t i c h c^n t i m 290 Ta V = V i + V2 - CO •2 V= 71 „ „ (x^fdx +n J2 1 V = V3 r3 (-3x + 10)2dx-7i ^ xMx + Tt C(9x^ 1 J2 \ n — 5 +100x) 1 I _ „ + 7n-2n 5 - 60x + 100)dx - 71 + 7i(3x^ - 3 0 x ^ SITI V = 5671 = 1 -3 l^dx f 3 dx - 1) - 71 (3 2 ,, (dvtt) 5 57o| T i n h the t i c h k h o i t r o n xoay tao nen k h i quay quanh true Ox h i n h S gidi h a n bdi y = xe",