Giới thiệu phương pháp tính tích phân và số phức: Phần 1

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Phần 1 tài liệu Phương pháp tính tích phân và số phức do Hà Văn Chương biên soạn cung cấp cho người đọc các kiến thức cơ bản, các bài tập ví dụ và bài tập tự ôn tập về họ nguyên hàm, tích phân xác định dành cho các bạn học sinh lớp 12 ôn tập lại kiến thức đã học. Mời các bạn tham khảo.

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Nội dung Text: Giới thiệu phương pháp tính tích phân và số phức: Phần 1

515.076 PH561P 1 iGV cbuyen Toan Trung tam luyen thi Vinh Viin - TP. HO Chi JViinhJ PHUONG PHAP TINH
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, c6 siia chita vd bo sung) IHU VIEN TiNH BIN'H THUAN NHA XUAT BAN DAI HOC QUOC GIA H A NOI
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 = f (x)dx ± g(x)dx c) kf(x)dx = k f(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 du = u + C 2. u"du = ^ — + C n + l fdu 3. = In u + C 4. e"du = e" + C u 5. cosudu = sinu + C 6. sinudu = -cosu + C f du 7. (1 + t a n u)du = t a n u + C cos^ u r du (1 + cot u)du = -cotu + C sin^ u du 1 , u - a = — I n + C. u^-a^ 2a u + a T] T i n h dao h a m cua F(x) = x.lnx - x, r o i suy ra nguyen h a m ciia f(x) = Inx. Gidi T a CO : F(x) = x . l n x - x = x(lnx - 1) 3
Suy r a : F'(x) = [x(lnx - 1)]' = Inx - 1 + - = Inx X 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 ra nguyen h a m ciia f(x) = 2xlnx. Gidi T a CO : F(x) = x^lnx nen F'(x) = 2xlnx + — .x^ = 2xlnx + x = f(x) + x Vay ' F'(x)dx= f (x)dx + dx 2 2 =:> ff(x)dx = F(x) - — + C = x^lnx - — + C J 2 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 + 1 biet F(0) = 2. Gidi Ta CO : f(x) = x V x + 1 = (x + 1 - l ) V x + 1 = (X + l ) V x + 1 - Vx + 1 = (X + 1)2 - (X + 1)2 3 1^ Vay f(x)dx = f{x + l ) 2 d x - ("(x + l ) 2 d x 3 1 (X + I ) 2 d ( x + 1 ) - f(X + l ) 2 d ( x + l ) 2 - 2 - = - ( X + l ) 2 - - ( X + 1)2 +C 5 3 = - ( x + l ) V x + l - - ( x + l)Vx + l + C Hay F(x) = - ( x + l ) V x + 1 - - ( x + 1) Vx + 1 + C 5 3 Vi F(0) = 2 nen t a CO : 2= - (0 + 1)^ Vo + 1 - - (0 + 1)V0 + 1 + C 5 3 5 3 15 V4y F ( x ) = - ( x + D^Vx + l - - ( x + l ) V x + l + — . 5 3 15 4
U 1 Cho F(x) = x h i x va g(x) = x^ I n ; x > 0. 1 . Chufng to r a n g : f(x) = - g \ x ) - - X . 2 2 2. Suy r a mot nguyen h a m F(x) ciia f(x). Gidi ^x^ 1 . Ta CO : g'(x) = 2 x l n + X (do X > 0) 4) Suy r a : 2xhi = g'(x) - X xhi = igXx)-ix Vay f(x) = - g ' ( x ) - - X (*) 2 2 2. Tix (*) t a suy r a : f(x)dx = - ("g'(x)dx - - xdx = - g ' ( x ) - — x^ + C = - x^ h i^1 + C J 2 J 2 J 2 4 2 4. Vay m o t nguyen h a m cua f(x) la : F(x) = - x I n v4y ~5\. Chufng m i n h r a n g F(x) = 9 + (x - 2 ) 6 " la mot nguyen h a m ciia {(x) = (x- l)e\ 2. Chufng m i n h r k n g G(x) = - ( 1 + 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 1. Ta CO : F'(x) = e" + e^Cx - 2 ) = e^lx - 1) = f(x) V a y F(x) l a m o t nguyen h a m cua f(x). 2. Ta CO : G(x) = - ( 1 + 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 = - ( 1 + x)e-'' + e " + C = - x e " + C V a y nguyen h a m cua k(x) \k K(x) = -xe"" + C. 5
~G\h dao hkm cua (p(x) = (ax + b)e''. Roi suy ra nguyen ham cua fix) = -xe". Gidi TiX gia thiet (p(x) = (ax + b)e'' Suy ra : (p'(x) = (a + ax + b)e'' De tinh nguyen ham ciia f(x) = -xe" ta chon a = - 1 , b = 1 Thi
2 a = - -5a = -2 5 12a - 3b = 6 b = -^ 5 6b - c = 0 c - - — el Chvlng m i n h F(x) = — I n X - a v 6 i a > 0 l a m o t nguyen h a m ciia 2a x + a f(x) = 1 v d i Vx ^ ± a. x^-a^ Gidi r ^x-a^ 2a 1 ^x + a j 1 (X + af T a CO : F'(x) = Vx 9^ ± a. 2a' rx-a^ 2a' ^x-a^ U + a, vx + a.j 2a X +a Suy r a : F'(x) = Vx ;t ± a 2a ( x + a f x-a 1 1 F'(x) = = flx) (x + a)(x - a) - a^ V a y F'(x) l a m o t nguyen h a m cua f(x) v d i Vx ^ ± a. neu x ^ O - I V 10 1 ChiJng m i n h F(x) = • X l a nguyen h a m cua 1 neu x = 0 (x-De^+l neu x^O f(x) = < x^ - neu x = 0 I 2 Gidi e^.x-Ce"-l).l (x-De^+l * Khix^Othi F'(x) = = f(x) X X V a y F(x) l a nguyen h a m cua f(x) t r e n (-oo, 0) u (0, + « ) (1) e" - 1 -1 * K h i x = 0 t h i F'(0) = l i m ^^^^ ^^^^ = l i m x-^'O X - 0 x 7
F'(0) = l i m ^ ^ = lim ^ X-.0 x^o 2x Vay FXO) = l i m — = - = fTO) (Quy t^c L'Hopital) (2) x^o 2 2 TiT (1), (2), t a suy r a F(x) l a nguyen h a m cua f(x) t r e n R. IT] T i n h dao h a m cua F(x) = (x^ - l ) l n 1 1 + x | - x^ln | x i . A2 Suy r a nguyen h a m cua f(x) = x l n 1+x Gidi Ta CO : F(x) = (x^ - l ) l n 11 + x | - x^ln | x I x^-1 Suy r a F'(x) = 2 x l n I x + 1 1 + x + 1 - 2xln I X I - = 2 x l n I X + 1 i - 1 - 2xln | x | v 6 i x 0, x ^ 1 f l + x^2 2 + x 1 + x 1 il = xln = 2xln Taco: f(x) = x l n l X X h x i - I n 1X1 ]= 2 x l n 1 1 + X 1- (**) Tir ( * ) , ( * * ) t a suy r a : f(x) = F'(x) + 1 Suy r a ff(x)dx = [F '(x)dx + f l d x = F(x) + x + C + X 1 V a y nguyen h a m cua f(x) = x l n I 1 la : F(x) = ( x 2 - D l n l l + x | - x ^ l n l x l + x + C. 12 I T i m a. b, c sao cho F(x) = e""^ (atan^x + btanx + c) l a m o t nguyen h a m cua f(x) = e'"^ .tan^x t r e n n n I 2'2 Gidi Taco : F'(x) = 7 2 . 6 " ^ ( a t a n ^ x + b t a n x + c) + e ' ' ^ [ 2 a ( l + tan^ x ) t a n x + b ( l + tan^ x)] 7t _ 71 v d i Vx e 2' 2. 7t 71 n 71 F(x) l a nguyen h a m cua f(x) t r e n F'(x) = f i x ) , Vx {'2' 2) '2' 2 8
e 2atan^ x + (yl2a + b ) t a n 2 x + (^^b + 2 a ) t a n x + (V2c + b) = e ' ' ^ t a n ^ x Vx e 2' 2 1 2a = 1 a = - 2 V2a + b = 0 b = - A ^ b + 2a = 0 2 . 1 V2c + b = 0 c =— 2 13 I Chutog m i i i h F(x) = | x | - l n ( l + I x I ) la mot nguyen ham cua fix) - 1+ Dai hoc Tong hap TP.HCM - 1993 Gidi x - l n ( l + x) neu x > 0 Ta CO : F(x) = 0 neu X = 0 - x - l n ( l - x) neu x < 0 neu x > 0 1 + x Ta CO : f(x) = 0 neu x = 0 V 1-x neu X < 0 Do do, t a CO : * Khi X > 0 thi F'(x) = 1 - = f(x) (1) 1+x 1+x * Khi x < 0 thi F'(x) = -1 + = f(x) (2) 1-x 1-x * vu- n.u^ l^vn+^ r F(x) - F(0> ,. x - l n ( l + x) * Ehi X = 0 thi F (0 ) = l i m = lim x^O" X - 0 x^O" X X ,. l n ( l + x) = lim lim x-»0* X x-»0* x Suy ra : F'(0*) = 1 - l i m ^— (do quy t i c L ' H o p i t a l ) x^.0^ 1+ X F'(O^) = 1 - 1 = 0
TV/r-.iu' v F(x) - F(0) ,. - X - ln(l - x) Mat khac : F (0 ) = l i m = lim x-»o- x - 0 X FXO-)=-l-liml^^^ x->0" X -1 ^ F'iOl = - 1 - l i m -i-tiL = - 1 + 1 = 0 x-»0- 1 (3) Vay F'(0*) = F'(0") = 0 Nen F'(0) = 0 = flO) TCr (1), (2), (3) suy ra F(x) la nguyen ham ciia f(x) tren R. — In x (X >0) 2 4 la nguyen ham cua 14 I Churngminh F(x) = 0 • (x = 0) fx In X (x > 0) f(x) 0 (x = 0) Dai hoc Yduac TP.HCM Gidi X X K h i X > 0 ta CO : F'(x) = x l n x + = xlnx = f(x) (1) 2 2 x^, x^ In X - Khi X = 0 ta CO : F(O^) = l i m ^^""^ ^^^^ = l i m ^ x^O* X - 0 x ^ O * X = l i m — In X - lim — = lim -0 x-^O* 2 x-yO* 4 x^.0* 2 = lim (quy t^c L'Hopital) x^o- 2 = lim = 0 = f(x) (2) x^o^V 2) Tix (1), (2) ta ket luan F(x) la nguyerI hkm cua fix) tren [0, +oo). fx'-2^ + 1] 15 i Tinh dao h^m cua ham so F(x) = In ^x^ + 2^R + lj x^-1 Roi suy ra ho nguyen ham cua f(x) = xUl 10
Gidi Ta CO : F(x) xac d i n h vdfi m o i x. x^ - 2^ +1 x^ + 2 A / X + 1 2V2(x2 - 1) x^ + 2V^ + 1 Ta CO : F'(x) = (x^ + 2V^ + 1)2 ' x^ - 2 A / i + 1 x^ + 2Vx + 1 2V2(x2 - 1) 2^{x'^ - 1) = 2V2f(x) x^ + 1 x^ - 2 7 ^ + 1^ Vay f(x)dx = ^ : F ( x ) = i l n 2A/2 2V2 x^ + 2A/X + 1 x^-l fx2-2>^ + l dx = In x^+1 x^ + 2Vx + 1 16 T i m ho nguyen h a m cua f(x) = max ( 1 , x^). Gidi 'x^ neu X < - 1 V 1 < X Ta CO : f i x ) = max ( 1 , x^) = 1 neu - 1 < X < 1 Vay: j x ^ d x neu x < - 1 v 1 < x + C neu x < - l v l < x m a x ( l , x2)dx = Idx neu - 1 < x < 1 X +C neu - 1 < X < 1 17 I T i m ho nguyen h a m ciia f i x ) = | l + x| - |l-x|. Gidi -2dx neu x < - 1 Ta CO : 1 +x - l +x dx = 2xdx neu - 1 < x < 1 2dx neu x > 1 -2x + C vdi X < -1 vay j ;1 + x - l +x dx = x^ + C vdi - 1 < X < 1 2x + C vdi X > 1 11
18 I T i m ho nguyen h a m ciia f(x) = | x | Gidi xdx neu x > 0 +C neu X > 0 Ta CO : I X I dx = -xdx neu x < 0 + C neu X < 0 2 19 I T i m ho nguyen h a m ciia f(x) = x i x |. Gidi x^dx neu x > 0 — +C neu X > 0 3 Ta CO : J x I x I dx = -x^dx neu x < 0 x^ + C neu x < 0 T i m ho nguyen h a m ciia f(x) = (x + \x\f. 20 Gidi Ta CO : (x+ I X I )2dx = Jlx^ + 2x I X I +x2 )dx ' 2 4x 4x dx = +C neu x > 0 O.dx = C neu X < 0 21 I T i m ho nguyen h a m ciia f(x) = cos X Dai hoc Yduac TP.HCM -2001 -He nhdn Gidi •COS xdx ' d(sin x) • d(sin x) T a CO : F(x) = f • cosx COS'^ X 1 - sin^ X sin^ X - 1 1, sin X - 1 -In + C. 2 I sin X + 1 22I T i m ho nguyen h a m cua fix) = 2\3^\5^\ Gidi 2250" Ta CO : f(x)dx = 2 \ 3 2 \ 5 3 M X = f(2.32.5^)''dx = + C. hi(2250) 12
23 I Tim ho nguyen ham cua : X* + 2x^ + X + 2 x^ + x^ + 1 a) f(x) = b) g(x) = X +x+1 X^ + X + 1 Dai hoc Ngoqi thuang - 1998 Gidi x^ + 2x2 + X + 2 x^ a) f(x)dx = dx = (x^ - X + 2)dx = 3 2 + 2x + C X + X + 1 x^ + x^ fx'' x" + X 1 C o x"^ X^ b) g(x)dx = — dx= ( x ^ - x + l)dx = + X + C. J x^ + X + 1 J 3 2 (hi X)' 24 I Tim ho nguyen ham cua f(x) = Gidi f(lnx)4 Ta CO : f (x)dx = dx = j ( h i x)"* d(ln x) = - (In x)^ + C. 25 \m ho nguyen ham cua f(x) = —— e" - 4e " DH Quoc gia Ha Ngi - D/1999 Gidi f e^dx d(e'') e" - 2 Ta CO : f(x)dx = =lln + C, 4 6^+2 e^^ Tim ho nguyen ham cua f(x) = e" + 1 Gidi Ta CO : '(e" + IKe^" - e" + 1) f (x)dx = ^-^dx = f ( e 2 ' ' - e ' ' + l ) d x = i e 2 ' ' - e ^ + x + C. +1 J 2 I 27 I Tim ho nguyen ham cua f(x) = 2x 1-e Gidi Ta CO : f eMx d(e'') die"") e" - 1 + C = - h i 6=^+1 f(x) = +C 1-e 2x 2 e" + 1 2 e" - 1 13
2 8 ] T i m ho nguyen h a m cua fix) = Ve" + e - 2. DH Y Thai Binh - 1997 Gidi Ta CO f(x) = Ve" + e"" - 2 = e2 - e 2 e2 - e 2 X X . ' X X e2 -e"2 neu —> — f(x) = 2 2 X X . ' X X -e2 + e 2 neu —< — 2 2 X X e2 - e'2 neu X > 0 f(x) = X X -e2 + e 2 neu X < 0 r X xA e2 + e 2 neu X > 0 Nen f (x)dx = e2 + e 2 neu X < 0 Inex 29 I T i m ho nguyen h a m ciia f(x) - 1 + xlnx HV Quan he Quoc te - 1997 Gidi T a CO : d ( l + x l n x ) = (1 + xlnx)'dx = ( l . l n x + — .x)dx X = (Inx + l ) d x = Inex.dx Inex.dx rd(l + xlnx) Vay f(x)dx = 1 + xlnx 1 + xlnx f (x)dx = I n 1 + x l n x + C. 3o] T i m ho nguyen h a m cua f(x) = x ( l - x)-°. DH Quoc gia Ha Ngi - 1998 Gidi Ta CO : fix) = [(x - 1) +1](1 - x)^" = (x - 1)^' + (x - 1)^°. 14
Nen f(x)dx= (x-l)2Mx + (x-l)2°dx (X-1)22 (x-l)21 (x-l)^M(x-l) + (x-l)2°d(x-l) = + C. 22 21 ,2001 31 I Tim ho nguyen ham cua f(x) = (1 + X2)1002 DH Quoc gia Ha Ngi - 2000 Gidi 1000 .2001 , 2000 f ..2 ^ Ta CO : f(x) = ( l + x^r^^ ( l + x ^ r o ' d + x^f 1 + x^ /- , NIOOO .2001 ' X J f (x)dx = dx = dx (1^^2)1002 1 + x^ •(l + x^)^ 2 \1000 2 ^ X ^ 1 + C. 1 + x' 1 + x^ 2002 1 + X ^ x^dx 32 1 Tinh tich phan — bang hai each bien ddi sau : (x^ + if a) Dat x = tana. 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 => dx = (tan^a + l)da, thi : x^^dx rtan^ a(tan^ a + l)da II = (x^ +1)^ (tan^ a +1)^ tan^ ada rsin^ a da (vi tan^a + 1 = — ^ — ) (tan^ a +1)^ cos a cos a cos'' a sin^ a. cos ada = sin ad(sin a) = — sin* a + C = - (tan* a. cos* a) + C 4 4 1 tan* a 1 x^ + C. 4 • (tan^ a + lf + C = -.- 4 ' (x^ + 1)^ 15
b) Dat u = + 1 => du = 2xdx r x^dx • x^.xdx 1 r(u - D d u I, (x^+l)^ ~ J (x^+l)^ '2 . -2 -3o.. 1 . 1 . l - 2 u , ^ - l ( l + 2x^) (u"" -u-')du = + — + Ci = — + Ci = — + C, 2u 4u^ 4u^ 4 (x^ + 1)2 ' 1 (1 + 2x2) T a xet : I i - I2 = + C+ - - C, 4(x2 + 1)2 4 ( l + x2)2 x^ + 2 x ^ + 1 I i - I2 = + C - Ci 4(x2 + i f I i - I2 - - + C - C i . 4 Vx^ + x"" + 2 33 I T i m ho nguyen h a m cua fix) = Gidi Vx^ +x-^ +2 V(x2+X-2)2 Ta CO : f(x)dx = ^ 3 4.x^ + 3 F(x) = xVx2 + 3 - F(x) + 31n(xVx2 + 3 ) + C Vay F(x) = i x V x 2 + 3 + - ln(xV(x2 + 3) + C. 2 2 16
35 I T i m ho nguyen h a m cua f i x ) = 1 + 8" HVNgdn hang -2000 Gidi dx 1 + 8" - 8" 'S^dx 8"dx Ta CO : F(x) = dx = Idx = x - 1 + 8" 1 + 8" 1 + 8" 1 + 8" du D a t u = 1 + 8" du = 8".ln8.dx = 8"dx. In 8 Nen _!L^x= f i i i L =J _ i n ( l ^ 8 " ) + C 1 + 8" J u In 8 In 8 dx Vay F(x) = = X - l n ( l + 8") + C. 1 + 8" hi 8 3 6 ) T i m ho nguyen h a m cua f(x) = Vx^ - x - 1 DH Y Thai Binh Gidi du dt D a t t = V u ^ + K + u => d t = : + 1 du Ap dung : i f 5 1 df 1^ d fx 1 + X — dx X r 2) 4 2 F(x) = - f Vx^ - x - 1 ^ ' 5 1 X + X — . 2J 4 2 F(x) = I n Vx^ - X - 1 + X - - + C. (x + 3)^ 37 \m ho n g u y e n h a m cua f i x ) = (x - 7)^ Gidi Dat u = x - 7 du = dx, u + 10 = x + 3 (x + 3 ) ' ^ ^ ^ j-(u_+10)' Vay F(x) = du (x-7)^ ^ u^ THLT VIcNTIf^HBiNHTHUAN 17
5-k k=0 k =0 5 -1-k 5 -1-k k =0 " k=0 -1-k + c. x^-l T i m ho nguyen h a m ciia f(x) = (x^ + 5x + IXx^ - 3x + 1)' DH Quoc gia Hd Ngi - A/2001 Giai x^-l 1 2x + 5 1 2x-3 Ta CO : • + - .- (x^ + 5x + IXx^ - 3x + 1) + 5x + 1 8 • - 3x + 1 2x + 5 J 1 2x - 3 , 1 , x-" - 3x + 1 F(x)= - - •dx + - -dx = - I n + C. 8 X +5x + 1 X -3x + 1 X + 5x + 1 39 i T i m ho nguyen h a m cua f(x) = x l n X. ln(ln x) Gidi Ta CO : (hi(lnx))' = xlnx dx (•[In(lnx)]' Nen f(x)dx = = In(lnx) + C. x l n x . h i ( l n x) h i ( l n x) (x + 1) 40 I T i m ho nguyen h a m cua f(x) = xQ + xe") Gidi Ta CO : (1 + xe")' = e^Cx + 1) (x + Ddx e^Cx + Ddx dCl + xe") f (x)dx = x d + xe'') xe^Cl + xe") [1 + xe^ - I J U + x e " ] [ ( l + x e ' ' ) - ( l + x e ' ' - D l d C l + xe") r d ( l + xe^ - 1) r d d + xe") [ l + x e " - l ] [ l + xe''] (1 + xe" - 1) d + xe") = hij 1 + xe^ - l l - Inl 1 + xe" I + C = In xe + C. 1 + xe^ 18
2x 41 I T i m ho nguyen h a m cua f(x) - x +Vx^ - 1 Gidi 2xdx f 2x(x - Vx^ - 1) Ta CO : f(x)dx = dx x +Vx^ - 1 x^ - (x^ - 1) 2x2dx - 2xVx2 - I d x = - x^ - (X^ -l)2d(x2 -1) 3 3 3 42 I T i m ho nguyen h a m ciia f(x) = Vx + 3 + Vx + 1 Gidi dx Vx + 3 - Vx - 1 Ta CO : f{x)dx = dx Vx + 3 + Vx + 1 J(x + 3 ) - ( x + l) i 1 f i (x + 3)2d(x + 3 ) - - (x + l)2d(x + l ) 2J = - V ( x + 3)=* - - V ( x + 1)^ +C. 3 3 3x + l A B 43 I 1. Xac d i n h cac h^ng so' A, B sao cho (X + 1)^ (x + 1)^ (x + 1)^ 3x + l 2. Dua vao k e t qua t r e n , t a t i m ho nguyen h a m cua f(x) = (X + ir Gidi 3x + l B Bx + (A + B) 1. Ta CO (x + if (x + 1)^ (x + if (x + If B = 3 B = 3 A + B =1 A = -2 3x + 1 -2 3 Vay (x + 1)^ (x + 1)^ (x + \f 3x + l dx dx 2. f(x)dx
3x^ + 3x + 3 44] Cho h k m so f(x) = x^ - 3x + 2 ' A B C 1. Xac d i n h cac h k n g so A, B, C de fix) = + +• (x - 1)2 x- 1 x +2 2. T i m nguyen h a m cua f(x). DHYDuac TP.HCM - 1996 Gidi 1. T a CO : x^ - 3x + 2 = (x - l)^(x + 2) Bx^ + 3x + 3 A B C Do do f(x) = + +• x^ - 3x + 2 (x - 1)2 X -1 x +2 o 3x^ + 3x + 3 = (B + Ox^ + (A + B - 2C)x + (2A - 2B + C) B +C =3 A =3 A + B - 2C = 3 B =2 2A - 2B + C = 3 C =1 3x2 + 3x + 3 2 1 2. f (x)dx =