Hãy kiên nhẫn học lại những điều rất cơ bản và làm cả những bài tập đơn giản. Chính kiến thức cơ bản giúp ta hiểu được những điều nâng cao sau này. Một vấn đề phức tạp là tổ hợp của nhiều vấn đề đơn giản, một bài toán khó là sự nối kết của nhiều bài toán đơn giản. Chỉ cần nắm vững vấn đề căn bản rồi bằng óc phân tích và tổng hợp chúng ta có thể giải quyết được rất nhiều bài toán khó.
515.076 PH561P iGV cbuyen Toan Trung tam luyen thi Vinh Viin - TP HO Chi JViinhJ PHUONG PHAP TINH MATH-EDUCARE HA VAN CHUCING (GV chuyen Toan Trung tarn luyen thi Vfnir Viin TP -Hd Chf Minh) PHl/CfNG PHAP T I N H TICK PHAN VA so PHUfC • LUYEN THI TU TAI VA DAI HOC m m m • CHirOfNG TRINH Mflfl NHAT CUA BO GIAO DUG VA DAO TAO • (Tdi ban idn thii nhat, IHU VIEN • c6 siia chita TiNH BIN'H • vd bo sung) THUAN NHA XUAT BAN DAI HOC QUOC GIA H A NOI www.matheducare.com MATH-EDUCARE TICH PHAN Hp N G U Y E N HAM K I E N THLfC C d B A N I D i n h nghia F(x) la nguyen hain cua fix) t r e n khoang (a; b) neu F'(x) = fix), Vx e (a; b) k i hieu F(x) = [f{x)dx II Tinh chat a) (Jf(x)dxj' = f(x) b) [ f (x) ± g(x)]dx = c) kf(x)dx = k f (x)dx ± f(x)dx g(x)dx k e R d) Neu F(t) la mot nguyen h a m ciia f(x) t h i F(u(x)) la mot nguyen ham ciia f [u(x)] u'(x) I I I B a n g n g u y e n h a m thi^dng d u n g v d i u = u ( x ) .n + l u"du = ^ — + C n + l = In u + C e"du = e" + C cosudu = sinu + C sinudu = -cosu + C du = u + C fdu u f du cos^ u r du sin^ u (1 + t a n u)du = t a n u + C (1 + cot u)du = -cotu + C du , u - a + C = — I n u^-a^ 2a u + a T] T i n h dao h a m cua F(x) = x.lnx - x, r o i suy nguyen h a m ciia f(x) = Inx Gidi T a CO : F(x) = x l n x - x = x(lnx - 1) www.matheducare.com www.matheducare.com Suy r a : F'(x) = [x(lnx - 1)]' = Inx - + - X = Inx Vay theo d i n h nghIa cua nguyen h a m , nguyen h a m ciia f(x) = Inx c h i n h la F(x) = x l n x - x + C ~2\h dao h a m ciia F(x) = x^lnx, r o i suy nguyen h a m ciia f(x) = 2xlnx Gidi T a CO : Vay ' F(x) = x^lnx nen F'(x)dx= F'(x) = 2xlnx + — x^ = 2xlnx + x = f(x) + x f (x)dx + dx 2 ff(x)dx = F(x) - — + C = x^lnx - — + C J 2 =:> V a y mo t nguyen h a m ciia f(x) la F(x) = x l l n x ~3\h nguyen h a m ciia f(x) - x V x + biet F(0) = Gidi Ta CO : f(x) = x V x + = (x + - l ) V x + = (X + l ) V x + - Vx + = (X + 1)2 - (X + 1)2 Vay f(x)dx = 1^ f{x + l ) d x - ("(x + l ) d x (X + I ) d ( x + ) - = - Hay Vi F(x) f(X + l ) d ( x + l ) - - ( X + l ) - - ( X + 1)2 +C MATH-EDUCARE - ( x + l ) V x + - - ( x + 1) Vx + + C = - ( x + l ) V x + l - - ( x + l)Vx + l + C = F(0) = nen t a CO : + - - (0 + 1)V0 + + C - (0 + 1)^ Vo 2= 3 15 F ( x ) = - ( x + D^Vx + l - - ( x + l ) V x + l + — 15 V4y MATH-EDUCARE U Cho F(x) = x h i x va Chufng to r a n g : f(x) g(x) = x^ I n = ; x > - g \ x ) - - X 2 Suy r a mot nguyen h a m F(x) ciia f(x) Gidi Ta CO : g'(x) = x l n ^x^ Suy r a : 2xhi Vay f(x) = - g ' ( x ) 4) (do + X = g'(x) - X > 0) xhi X - - X = igXx)-ix (*) Tix (*) t a suy r a : J f(x)dx = - ("g'(x)dx - J J xdx = - g ' ( x ) - — x^ + C = - x^ h i^1 4 Vay m o t nguyen h a m cua f(x) la : F(x) = - x I n + C v4y ~5\ Chufng m i n h r a n g F(x) = + (x - ) " la mot nguyen h a m ciia {(x) = (x- l)e\ Chufng m i n h r k n g G(x) = - ( + x)e"'' la m o t nguyen h a m cua g(x) = x.e'" Roi suy r a nguyen h a m cua k(x) = (x - De"" Gidi Ta CO : F'(x) = e" + e^Cx - ) = e^lx - 1) = f(x) V a y F(x) l a m o t nguyen h a m cua f(x) Ta CO : G(x) = - ( + x)e"'' Suy r a : G'(x) = -e"" + (1 + x)e-'' = e"\ = g(x) Vay G(x) la mot nguyen h a m ciia g(x) Suy r a nguyen h a m cua k(x) = (x - l)e~'' = xe"" - e"'' = g(x) - e" Nen k(x)dx = Jg(x)dx - je-^dx = G(x) + e"" + C = - ( + x)e-'' + e " + C = - x e " + C V a y nguyen h a m cua k(x) \k K(x) = -xe"" + C www.matheducare.com www.matheducare.com ~G\h dao hkm cua (p(x) = (ax + b)e'' Roi suy nguyen ham cua fix) = -xe" Gidi (p'(x) = (a + ax + b)e'' Suy : (p(x) = (ax + b)e'' TiX gia thiet De tinh nguyen ham ciia f(x) = -xe" ta chon a = - , b = Thi ^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 : s s: - i = >^ 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 MATH-EDUCARE c ) T a CO : {S (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 MATH-EDUCARE 10 32i (1 + i) Nen (V3 + i ) ^ -512i Vay p h a n thuc l a d)Tac6: , p h a n ao l a 16 z + - = l z 16' 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 www.matheducare.com www.matheducare.com 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 lisl 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 MATH-EDUCARE 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 MATH-EDUCARE 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 MATH-EDUCARE 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 MATH-EDUCARE 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 www.matheducare.com www.matheducare.com = -2cos- + g) 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 — > + MATH-EDUCARE + *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 MATH-EDUCARE 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 www.matheducare.com www.matheducare.com D e (C) • e (C) C • B e (C) • A e (C) o o » L a y ( ) - ( ) t a CO : (4 - a)2 + (3 + (2-af V3 + {3 + S - b)' = R ' - hf = R^ (1) (2) (4) (3 - af + ( - b)^ = R ^ (3) ( - a ) ' + (3 - b f = R ' 12 - a = => T h e a = 3, b = v a o ( ) =o T h e a = v a o (1) v a ( ) « 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) (5 + i ) * MATH-EDUCARE T a CO : nen 476 + 480i 239+ i ma 239+ i 239+ i (5 + i)'^ (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 MATH-EDUCARE 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 www.matheducare.com www.matheducare.com 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 ^ cp^ + ism va MATH-EDUCARE b) T a CO : 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 cos = cos sincp - icoscp - cos - - - C P f57I u isin 2y + isin fn + isin " - ! f5n u ^-2 396 MATH-EDUCARE 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 www.matheducare.com www.matheducare.com 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 K I E N THtfC CO BAN 39 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 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 K I E N THtfC CO B A N 326 MATH-EDUCARE 328 BAI TAP 398 y y y y X y y MATH-EDUCARE •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 [...]... x^ 1 + x' ^ 1 + x^ 1 + x^ ^ 32 1 Tinh tich phan , ' dx = 2 2002 1 + C + X ^ x^dx — bang hai each bien ddi sau : (x^ + if b) Dat u = x^ + 1 So sanh hai ket qua t i m ducfc Dai hoc Tong hap TP.HCM ~ A/1977 Gidi Ta CO a) Dat : X = tana II = => dx = (tan^a + l)da, x^^dx rtan^ a(tan^ a + l)da (x^ +1)^ (tan^ a +1)^ tan^ ada rsin^ a (tan^ a +1)^ cos a sin^ a cos ada = = thi : da (vi tan^a + 1 = — ^ — ) cos... 2\3^\5^\ Gidi Ta CO : f(x)dx = 2 \ 3 2 \ 5 3 M X = f(2.32.5^)''dx = 2250" hi(2250) + C 12 MATH-EDUCARE 23 I Tim ho nguyen ham cua : X* + a) f(x) = X 2x^ + X + 2 b) g(x) = +x+1 x^ + x^ + 1 X^ + X + 1 Dai hoc Ngoqi thuang - 1998 Gidi f(x)dx = a) x^ + 2x2 X + X + + X + 2 1 x^ + x^ 1 fx'' x" + X g(x)dx = — dx= J x^ + X + 1 b) dx = (x^ - X + x^ 2)dx = 3 C o x"^ X^ 3 2 ( x ^ - x + l)dx = J (hi 24 I Tim ho... ciia f(x) = (x + — +C 3 neu X > 0 x^ + C neu x < 0 \x\f Gidi Ta CO : (x+ I X I )2dx = Jlx^ + 2x I X I +x2 )dx ' 4x 2 4x dx = +C O.dx = C 21 I T i m ho nguyen h a m ciia f(x) = neu x > 0 neu X < 0 cos X Dai hoc Yduac TP.HCM -2001 -He nhdn Gidi MATH-EDUCARE T a CO : F(x) ' d(sin x) •COS xdx = f • cosx • d(sin x) sin^ X - 1 1 - sin^ X COS'^ X 1, sin X - 1 + C -In 2 22I I sin X + 1 T i m ho nguyen h a m cua... 34 + C= = ln(x.Vx2 + 3 ) + C T i n h F(x) = Inixl 4x^ -1999 Gidi = [Vx2 + 3dx = xVx2 + 3 - jx.d(Vx2 + 3) Fix) = x Vx^ + 3 - +C fVx2 + 3dx DHYHaNoi MATH-EDUCARE ^•^'^^ > ^ 3 = x V 7 T i - (Tich p h a n t i i n g phan) f V 7 7 ^ d x + 3'" 4.x^ + 3 F(x) = xVx2 + 3 - F(x) + 31n(xVx2 + 3 ) + C F(x) = i x V x 2 + 3 + - ln(xV(x2 + 3) + C 2 2 Vay 16 MATH-EDUCARE 35 I T i m ho nguyen h a m cua f i x ) = 1... - 2 dx (X + If dx + 3 (x + If = - 2 (x + ir^dx + 3 (x + ir'^dj^ = (X+ i r - 3 ( x + i r + c = (x + lf X + 1 + c 19 www.matheducare.com www.matheducare.com 44] Cho h k m so f(x) = 3x^ + 3x + 3 x^ - 3x + 2 ' 1 Xac d i n h cac h k n g so A, B, C de fix) = A (x - 1)2 B + C +• x +2 x- 1 2 T i m nguyen h a m cua f(x) DHYDuac TP.HCM - 1996 Gidi 1 T a CO : Do do x^ - 3x + 2 = (x - l)^(x + 2) f(x) = o (x -... J cos x 56 I Tim ho nguyen ham cua ham so f(x) = cos X + s i n x cos x 2 + sin X Gidi Ta CO : f(x) = cos x + sin X cos x _ 2 cos x + sin x cos x - cos x 2 + sin x 2 + sin x cos x(2 + sin x) - cos x 2 + sin X F(x) = fcos x +sin X cosx 2 + sin X = cos dx = X - cos x 2 + sin X cos X V COSX - 2 + sin X, dx = sinx - ln(2 + sinx) + C 57 I Tim ho nguyen ham cua ham so f(x) = sin 3x sin 4x tan X + cot2x Gidi... Ta CO : Vx + 3 - Vx - 1 dx J(x + 3 ) - ( x + l) dx f{x)dx = Vx + 3 + Vx + 1 i 1 f i (x + 3)2d(x + 3 ) - - (x + l)2d(x + l ) 2J = - V ( x + 3)=* - - V ( x + 1)^ 3 3 +C 3x + l 43 I 1 Xac d i n h cac h^ng so' A, B sao cho (X + 1)^ A B (x + 1)^ (x + 1)^ 2 Dua vao k e t qua t r e n , t a t i m ho nguyen h a m cua f(x) = 3x + l (X + ir Gidi 1 Ta CO Vay 2 B Bx + (A + B) (x + if (x + If 3x + l (x + 1)^ (x +... sin9x + sinx] 4 MATH-EDUCARE Vay F(x)= f (x)dx = 1 4 J f sin3xsin4x tan X + cot 2x dx (sin 5x + sin 3x - sin 9x + sin x)dx 1 - — cos 5x - — cos 3x + — cos 9x - cos x+ C 5 3 9 4 Tim ho nguyen ham cua ham so' f(x) (sin''x + cos'*x)(sin®x + cos^x) HV Quan he Quoc te -D/1997 24 MATH-EDUCARE Gidi Ta CO : f(x) = 1 1- 9 - sin^ 2 2x \ 3 1- - 5 2x sin^ = 1 4 5 1 = 1 — - (1 - cos 4 2 3 s i n ^ 2x + - sin"* 2x... 2 2 = — [cos4x - cos8x + cos6x - cos2x] 4 Vay G(x) = — (cos 4x - cos 8x + cos 6x - cos 2x) dx 4 J — sin 4x - — sin 8x + — sin 6x - — sin 2x+ C 4 8 6 2 MATH-EDUCARE 63 sin X Tim ho nguyen ham ciia ham so f(x) = 1 + sin 2x : Ta CO f(x) = Gidi sin sin X l + sin2x t=x-— 4 Dat DH Bach khoa Ha Noi sin X o^^^sf^ ^X X (sinx + cosx)^ Z COS o x = t+ — 4 o -1 4 dx = dt 26 MATH-EDUCARE sin f(x) = Nen 2 cos^ t... 3A + B = 0 -5A - B = 1 A = - i 2 B =^ 2 r dt 3 — ^ =- i l n | t - l | + iln|3t-5i +C F(x) = - 2 2 t - 1 + -2 J3t - 5 3t-5 3 tan X - 5 1 =i l n + C = - In + C 2 2 t-1 tan X - 1 Tim ho nguyen ham cua ham so f(x) = COS X COS 71 X + — I 4; Gidi Ta CO : f(x) = COS X COS 71 X + — C O S x(cos X - sin x) 4 cos x cos^ x(cos X - sin x) Vay F(x) = f (x)dx = A/2 cos^ x 1 - tan x dx 1 cos^ x 1 - tan X = V2 f —