Phương pháp tính tích phân và số phức-hà văn chương

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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, IHU VIEN • c6 siia chita TiNH BIN'H • vd bo sung) 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 = 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 ) 2. .n + l u"du = ^ — + C n + l = In u + C 4. e"du = e" + C cosudu = sinu + C 6. sinudu = -cosu + C du = u + C 3. fdu u 5. 7. 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 1 , 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 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 + - 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 ra 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 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 Vay f(x)dx = 1^ f{x + l ) 2 d x - ("(x + l ) 2 d x 3 1 (X + I ) 2 d ( x + 1 ) - 2 = - 2 F(x) f(X + l ) 2 d ( x + l ) - - ( X + l ) 2 - - ( X + 1)2 5 Hay Vi +C 3 - ( x + l ) V x + 1 - - ( x + 1) Vx + 1 + C 5 3 = - ( x + l ) V x + l - - ( x + l)Vx + l + C = F(0) = 2 nen t a CO : + 1 - - (0 + 1)V0 + 1 + C - (0 + 1)^ Vo 5 2= 5 3 3 15 F ( x ) = - ( x + D^Vx + l - - ( x + l ) V x + l + — . 5 3 15 V4y 4 U 1 Cho F(x) = x h i x va 1 . Chufng to r a n g : f(x) g(x) = x^ I n = ; x > 0. - g \ x ) - - X . 2 2 2. Suy r a mot nguyen h a m F(x) ciia f(x). Gidi 1 . Ta CO : g'(x) = 2 x l n ^x^ Suy r a : 2xhi Vay f(x) = - g ' ( x ) 2 4) (do + X = g'(x) - X > 0) xhi X - - X 2 = igXx)-ix (*) 2. Tix (*) t a suy r a : J f(x)dx = - ("g'(x)dx - 2 J 2 J xdx = - g ' ( x ) - — x^ + C = - x^ h i^1 2 4 2 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) = 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 (p'(x) = (a + ax + b)e'' Suy ra : (p(x) = (ax + b)e'' TiX gia thiet De tinh nguyen ham ciia f(x) = -xe" ta chon a = - 1 , b = 1 Thi (p'(x) = -xe" Vay nguyen ham ciia f\x) = -xe" la F(x) = (-x + De" + C. ~T\g minh F(x) = In| x + Vx^ + K | la mot nguyen ham cua fix) = , ^ Vx^+K tren R. Gidi ^ , i a CO : 1+ . (X + V 7 7 K ) ' r (x) = X + Vx^ + K Suy ra : F'(x) = , ^ X VX2 + K x + Vx^ + K VX^ + K + Vx^ + K ( x + Vx^ + K ) = f(x) Vx e R. Vx2+K Do do : F(x) la mot nguyen ham cua f(x) tren R. sl Cho ham so fix) = xV3 - x vdi x < 3. Tim cac so' a, b sao cho ham so' F(x) = (ax^ + bx + c) V3 - x la mot nguyen ham cua f(x). Dai hoc Su pham Ki thudt TP.HCM Gidi Ta CO : F'(x) = (2ax + b) V 3 - x (ax^ + bx + c). 2V3-X 1 2V3-X . . .2 [(2ax + b).2(3-x) - (ax' + bx + c)] 1 . [-5ax^ + (12a - 3b)x + 6b - c)] 2A/3-X F(x) \k mot nguyen hkm cua f(x) k h i x < 3 -5ax^ + (12a - 3b)x + 6b - c = 2(3 - x)x o F'(x) = f(x) o Vx < 3 Vx < 3 6 2 a = 5 -5a = -2 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 x + a 2a 1 x^-a^ f(x) = v d i Vx ^ ± a. Gidi r ^x-a^ T a CO : F'(x) = 1 ^x + a j F'(x) = 1 2a' rx-a^ U Suy r a : F'(x) = 2a af (X + Vx ^x-a^ 2a' 9^ ± a. vx + a.j + a, 2a +a X 2a ( x + a f Vx ;t ± a x-a 1 1 (x + a)(x - a) - a^ = flx) V a y F'(x) l a m o t nguyen h a m cua f(x) v d i Vx ^ ± a. 10 1 ChiJng m i n h F(x) = • neu x ^ O X 1 neu x = 0 (x-De^+l x^ f(x) = < - I I V l a nguyen h a m cua neu x^O neu x = 0 2 Gidi * Khix^Othi F'(x) = e^.x-Ce"-l).l (x-De^+l X = f(x) X V a y F(x) l a nguyen h a m cua f(x) t r e n (-oo, 0) u (0, + « ) e" - 1 * K h i x = 0 t h i F'(0) = l i m ^^^^ x-^'O ^^^^ = l i m X - 0 (1) -1 x 7 IT] Vay F'(0) = l i m X-.0 ^ ^ = lim x^o FXO) = l i m — = - = fTO) x^o 2 2 2x ^ (Quy t^c L'Hopital) (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. T i n h dao h a m cua F(x) = (x^ - l ) l n 1 1 + x | - x^ln | x i . Suy r a nguyen h a m cua f(x) = x l n 1+x A2 Gidi Ta CO : Suy r a Taco: F(x) = (x^ - l ) l n 11 + x | - x^ln | x I x^-1 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 2 f l + x^2 + x + x il = xln = 2xln f(x) = x l n l X X 1 h x 1 i - I n 1X1 ]= 2 x l n 1 1 + X f(x) = F'(x) + 1 Tir ( * ) , ( * * ) t a suy r a : Suy r a 1- (**) ff(x)dx = [F '(x)dx + f l d x = F(x) + x + C V a y nguyen h a m cua f(x) = x l n I 1 + X 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 n n cua f(x) = e'"^ .tan^x t r e 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)] v d i Vx e 7t F(x) l a nguyen h a m cua f(x) t r e n 71 {'2' 2) <=> F'(x) = f i x ) , Vx 7t _ 71 2' 2. n 71 '2' 2 8 2atan^ x + (yl2a + b ) t a n 2 x + (^^b + 2 a ) t a n x + (V2c + b) = e ' ' ^ t a n ^ x <=> e Vx e 2' 2 1 a = 2 2a = 1 V2a + b = 0 b = - A ^ b + 2a = 0 2 . 1 c =— 2 V2c + b = 0 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) Ta CO : F(x) = neu x > 0 0 neu - x - l n ( l - x) 0 X = neu x < 0 neu x > 0 Ta CO : f(x) = 1 + x 0 neu x = 0 V 1-x Do do, t a * Khi neu CO : 0 thi F'(x) = 1 - * Khi x < 0 thi F'(x) = -1 + * vu* Ehi X > X = n.u^ 0 thi 1+x 1+x 1-x = f(x) 1-x (1) = f(x) (2) l^vn+^ r F(x) - F(0> ,. x - l n ( l + x) F (0 ) = l i m = lim X - 0 x^O" = lim x-»0* F'(0*) = 1 - l i m x^.0^ F'(O^) = 1 - 1 x^O" X ,. l n ( l + x) lim X x-»0* X Suy ra : 0 X < ^— 1+ X = 0 x (do quy t i c L ' H o p i t a l ) TV/r-.iu' v F(x) - F(0) Mat khac : F (0 ) = l i m ,. - X - ln(l - x) = lim x - 0 x-»o- X FXO-)=-l-liml^^^ x->0" X -1 ^ F'iOl = - 1 - l i m -i-tiL = - 1 + 1 = 0 1 x-»0- Vay F'(0*) = F'(0") = 0 (3) F'(0) = 0 = flO) Nen TCr (1), (2), (3) suy ra F(x) la nguyen ham ciia f(x) tren R. 14 I Churngminh F(x) = — In x (X 2 4 • 0 fx In X f(x) >0) la nguyen ham cua (x = 0) (x > 0) (x = 0) 0 Dai hoc Yduac TP.HCM Gidi K h i X > 0 ta CO : F'(x) = x l n x + 2 2 X X (1) = xlnx = f(x) x^, In Khi 0 ta X = CO X - : F(O^) = l i m ^^""^ ^^^^ = l i m X - 0 x^O* = l i m — In X x-yO* 4 x^ X -0 2 x^.0* (quy t^c L'Hopital) 2 x^o- ^ x ^ O * - lim — = lim 2 x-^O* = lim = lim = 0 = f(x) x^o^V (2) 2) Tix (1), (2) ta ket luan F(x) la nguyerI hkm cua fix) tren [0, +oo). 15 i Tinh dao h^m cua ham so F(x) = In fx'-2^ + 1] ^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^ Ta CO : F'(x) = +1 2V2(x2 - 1) x^ + 2 A / X + 1 x^ + 2V^ + 1 (x^ + 2V^ + 1)2 ' x^ - 2 A / i + 1 x^ + 2Vx + 1 2V2(x2 - 1) 2^{x'^ - 1) = 2V2f(x) x^ + 1 f(x)dx = Vay x^-l x^+1 16 x^ - 2 7 ^ + 1^ ^ : F ( x ) = i l n 2A/2 2V2 x^ + 2A/X + 1 dx = In fx2-2>^ + l x^ + 2Vx + 1 T i m ho nguyen h a m cua f(x) = max ( 1 , x^). Gidi Ta CO : f i x ) = max ( 1 , x^) = 'x^ neu 1 X < neu - 1 - 1 V 1 < X < < X 1 Vay: j x ^ d x neu x < - 1 v 1 < x + C neu m a x ( l , x2)dx = Idx neu - 1 < x < 1 17 I T i m ho nguyen h a m ciia f i x ) = | l + x| - X +C x < - l v l < x neu - 1 < X < 1 |l-x|. Gidi Ta CO : vay 1 +x - j ;1 + x - l +x dx = -2dx neu x < - 1 2xdx neu - 1 < x < 1 2dx neu x > 1 -2x + C vdi l +x dx = x^ + C 2x + C X < -1 vdi - 1 < X < vdi 1 X > 1 11 18 I T i m ho nguyen h a m ciia f(x) = | x | Gidi xdx Ta CO : neu x > 0 +C neu X > 0 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 Ta CO : neu x > 0 J x I x I dx = -x^dx 20 — +C 3 neu X > 0 x^ neu x < 0 T i m ho nguyen h a m ciia f(x) = (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 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 fix) = 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 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 nguyen ham cua f(x) = 2 + 2x + C + X + C. X)' Gidi Ta f (x)dx = CO : f(lnx)4 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 Ta f(x)dx = CO : f e^dx Tim ho nguyen ham cua f(x) = d(e'') e" - 2 =lln + C, 4 6^+2 e^^ e" + 1 Gidi Ta CO : f (x)dx = '(e" + IKe^" - e" + 1) ^-^dx +1 I 27 I Tim ho nguyen ham cua f(x) = = 1-e f ( e 2 ' ' - e ' ' + l ) d x = i e 2 ' ' - e ^ + x + C. J 2 2x Gidi Ta f(x) = CO : f eMx 1-e 2x d(e'') die"") 2 e" - 1 + C = - h i 6=^+1 +C 2 e" + 1 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 = -e"2 e2 X X f(x) = X e2 - e 2 . neu ' X X —> 2 e2 - e 2 — 2 X -e2 + e 2 X f(x) = . neu ' X X —< 2 — 2 X e2 - e'2 X neu 0 X > X -e2 + e 2 r neu 0 X < xA X e2 + e 2 Nen neu 0 X > f (x)dx = e2 + e 2 29 I T i m ho nguyen h a m ciia f(x) - neu 0 X < Inex 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 Vay f(x)dx = 1 + xlnx 1 + xlnx rd(l + xlnx) Inex.dx 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 f(x)dx= Nen (x-l)2Mx + (x-l)^M(x-l) + (x-l)2°dx (x-l)2°d(x-l) = (X-1)22 (x-l)21 22 21 + C. ,2001 31 I Tim ho nguyen ham cua f(x) = (1 + X2)1002 DH Quoc gia Ha Ngi - 2000 Gidi .2001 Ta CO : f(x) = , ( l + x^r^^ f 2000 ( l + x ^ r o ' d + x^f /- .2001 J f (x)dx = (1^^2)1002 2 a) Dat x = tana. X 1000 NIOOO X •(l + x^)^ \1000 dx 2 1 ^ 1 + 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 a cos'' a sin ad(sin a) — sin* a + C = - (tan* a. cos* a) + C 4 4 tan* a 1 x^ 1 + C. + C = -.4 • (tan^ a + lf 4 ' (x^ + 1)^ 15 b) Dat I, u = r + 1 x^dx => du = 2xdx • x^.xdx (x^+l)^ ~ J (x^+l)^ -2 -3o.. (u"" -u-')du = 1 r(u - D d u '2 . 1 . 1 . l - 2 u , ^ - l ( l + 2x^) + — + Ci = — + Ci = — + C, 4u^ 4 (x^ + 1)2 ' 4u^ 2u T a xet : I i - I2 = 4(x2 + 1)2 I i - I2 = 1 (1 + 2x2) + C+ - C, 4 ( l + x2)2 x^ + 2 x ^ + 1 4(x2 + i f + C - Ci I i - I2 - - + C - C i . 4 33 I T i m ho nguyen h a m cua fix) = Vx^ + x"" + 2 Gidi Ta CO : Vx^ +x-^ f(x)dx = I x2 + X-2 I 0) 1 X-* + =hilxl - — 4 X 34 + C= = ln(x.Vx2 + 3 ) + C . T i n h F(x) = Inixl 4x^ +C. fVx2 + 3dx . DHYHaNoi -1999 Gidi = [Vx2 + 3dx = xVx2 + 3 - jx.d(Vx2 + 3) Fix) = x Vx^ + 3 - ^•^'^^ > ^ 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 35 I T i m ho nguyen h a m cua f i x ) = 1 + 8" HVNgdn hang -2000 Gidi Ta CO : dx F(x) = 1 + 8" - 8" dx = 1 + 8" 1 + 8" Vay du In 8 = x - 8"dx 1 + 8" = 8"dx. f i i i L =J _ i n ( l ^ 8 " ) + C J u In 8 In 8 _!L^x= 1 + 8" dx F(x) = 1 + 8" du = 8".ln8.dx D a t u = 1 + 8" Nen 'S^dx Idx 1 + 8" = X - l n ( l + 8") + C. hi 8 3 6 ) T i m ho nguyen h a m cua f(x) = Vx^ - x - 1 DH Y Thai Binh Gidi D a t t = V u ^ + K + u => d t = : du + 1 du dt Ap dung : df dx F(x) = Vx^ - x - 1 d 1^ X - f i fx r ^ . F(x) = I n Vx^ - X - 1 + X - X f 1 2) 5 4 ' 5 2J 4 + X + X 1 — 2 1 — 2 + 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 Vay F(x) = du = dx, u + 10 = x + 3 (x + 3 ) ' ^ ^ ^ j-(u_+10)' (x-7)^ ^ u^ du THLT VIcNTIf^HBiNHTHUAN 17 5-k k=0 k =0 " k =0 -1-k 5 T i m ho nguyen h a m ciia f(x) = 5 k=0 -1-k -1-k + c. x^-l (x^ + 5x + IXx^ - 3x + 1)' DH Quoc gia Hd Ngi - A/2001 Giai Ta CO : F(x)= x^-l 2x + 5 1 2x-3 • + - .+ 5x + 1 8 • - 3x + 1 1 (x^ + 5x + IXx^ - 3x + 1) - 8 X 2x + 5 J 1 •dx + 5x + 1 + 39 i T i m ho nguyen h a m cua f(x) = X 2x - 3 , 1 , x-" - 3x + 1 -dx = - I n 3x + 1 X + 5x + 1 + C. - x l n X. ln(ln x) Gidi Ta CO : (hi(lnx))' = Nen xlnx (•[In(lnx)]' dx f(x)dx = x l n x . h i ( l n x) h i ( l n x) = In(lnx) + C. (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) f (x)dx = xe^Cl + xe") x d + xe'') e^Cx + Ddx (x + Ddx dCl + xe") [1 + xe^ - I J U + x e " ] (1 + xe" - 1) [ l + x e " - l ] [ l + xe''] r d ( l + xe^ - 1) [ ( l + x e ' ' ) - ( l + x e ' ' - D l d C l + xe") = hij 1 + xe^ - l l - Inl 1 + xe" I + C = In xe 1 + xe^ r d d + xe") d + xe") + C. 18 41 I T i m ho nguyen h a m cua f(x) - 2x x +Vx^ - 1 Gidi Ta CO : f(x)dx = 2xdx f 2x(x - Vx^ - 1) x +Vx^ - 1 x^ - (x^ - 1) 2x2dx - 3 2xVx2 - I d x = - x^ 3 dx (X^ -l)2d(x2 -1) 3 42 I T i m ho nguyen h a m ciia f(x) = Vx + 3 + Vx + 1 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 + if B = 3 B = 3 A + B =1 A = -2 3x + 1 -2 3 (x + 1)^ (x + 1)^ (x + \f 3x + l f(x)dx (X + If - Xem thêm -