Phương pháp tính tích phân và số phức phần 1

515.076 PH561P 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 ) .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) 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 + ) - = - F(x) f(X + l ) d ( x + l ) - - ( X + l ) - - ( X + 1)2 Hay Vi +C - ( 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 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 ~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 t^ = - x^ => tdt = - xdx 121 267 D o i can Vay 1= I = Tinh I = t =1 X = t = X = f1 Cx^yll - x^dx = f ^ x ^ V l - x ^ x d x = rl '\t2 - t ' ' ) d t = JO rl f (l-t^H.C-tdt) 15 [3 " s j 2^ ft' Vl-x^dx Gidi Dat x = sint I = Vay I = Vay dx - costdt vi t e "1 71 x = l D o i can X = f V l - x^dx = f Vl - sin^ t cos t d t = cos^ t d t t = t = 0 I cos t| costdt t h i cost > _ 7t - + cos 2t dt = — t + — sin 2t ^4 268| T i n h I = 'VT^dx^^ f ^ x V l - xdx DH Y duac TP.HCM - 1997 Gidi I = Vay I Ta CO : ^ x V l - xdx = [ ^ [ ( x - l ) + l ] V l - x d x t^dt vdi t = t2dt- ,1 ^2 (t2 - t ) d t = - t Jo - X -it2 _ x V l - xdx = — 122 269 Tinh I = DH Thuy Igi - 1997 Gidi Dat X = 2sint Doi can Vay dx = 2costdt X = 2 X = I = 270 1= Tinh I = - x = 4(1 - Sinn) = c o s n sin^ t.2 cos t.2 cos tdt I = sin^ 2tdt = Vay va t = 2(l-cos4t)dt = t - — s i n 4t = n f x^ V4 - x^dx - n f2V3 dx DH Kiwi A - 2003 Gidi T a CO : I = ^ Dat dx 2V3 2V3 xVx2+4 -"^ xdx x^VZTI t2 - = x^ t = Vx^ + => dt = xdx Vx^+4 Doi can I = X = 2V3 X = V5 =:> t = r" dt t2 - t = -In 44 t -2 ^ t + 3^4 n in - - In - = — In V 5j ( Gtdi Dat X = cost, < X < — n e n ta chon —< t < — 123 272 X = K h i t a c6 V2 X = I = = t = => - ^ va dx = - s i n t d t n t = — f T l + COSt •7 l l + cos t sS iI n t d t = - ^1 Sin t d t sint V1 - COS t - = -14 Vay Tinh I = (1 + cos t ) d t = - ( t + s i n t) Vl-x X + ^ ^x + dx = - + 4 7r , = - + A/2 2 dx 6t - A/2005 Gidi D a t t = VxTT =>t^ = x + l = > x= t ^ - l = > x + = t^ + l => d x = 3t^dt Doi can I = I273I t = t = x = X = x + *^x + l f2(t^ + l ) t ^ dx = f2 dt = h i (t^ + t ) d t = ft' [5^ t'' 2) 231 10 dx Tinh I = Dii bi - B/2008 Gidi Dat V4 - t = Doi can x^ t^ = X^ - t = Vs X = t = X = •1 I = x^ dx = - Jo -> xdx = - t d t r/3(4-t2)tdt r = 4tI t(4-t2)dt V3 J2 t^i — 3^ V3 1^-3V3 124 74 Tinh I = •1 2x^ - 2x + dx •"^ Vx^ - 2x + Gidi Xet ham F(x) = (ax + b)Vx^ - x + 2, ta CO : X n — (ax + bXx - D 2ax^ + (b - 3a)x + 2a - b F (x) = aVx - 2x + + = Vx^ - 2x + Vx^ - 2x + T-v 2x^ - 2x + 2ax^ + (b - 3a)x + 2a - b Vx^ - 2x + Vx^ - 2x + —= Cho F(x) = (x + Vay 1= r I DVX^ ^f" , ta a = 1, b = 2x + - -9.^ ^ ^ Hv = ( V ^ ^9.' * Vx^ - 2x + 275| T i n h I = CO =9.-^ » V2x + + DHkhoi D -2011 Gidi Dat t = V2x + + => dt = 2V2x + V2x + V x + Idt = dx Ta CO : Doi can Vay 1= t - = V2x + o X = t = X = t = [-4 "3 4x-l V2x + + dx = dx J :dx = o (t - 2)dt = dx 2t^ - 8t + = 4x - •6 (2t^ - 8t + 5)(t - 2)dt t 2t2 - t + - ^ j d t = 2t^ -6t^ +21t-101nt 3 Tinh I = dx Da bi -D/2008 125 I = Jo xe 2x X - dx = Gidi •0 xe^^dx + 'xd(e2'') = i xe^^ Jo =-(e" = -e'^^{2x-li l2 = Vay I •1 - x d x Jo -t Jo V4x + ^ •1 - x d x Jo Jo +1) =A/3-2 = ^|4-x^ xe^" - •2 x + 277) T i n h I = dx = V3 + 6^-7 dx Gidi Bat t = V4x + D o i can X = X = => = 4x + => X dx = = tdt t =1 t = •2 x + I = Jo V x + 278| T i n h I = dx = fSt^ Ji 't' + dt 24 16 n 3t [^Vx^ + I d x Gidi Dat u = Vx^ + => = x^ + 2xdx du = 2udu = 2xdx dv = dx V = ^ X 2Vx2 + Ta CO : I = f W x ^ + I d x = xVx^ + Jo :.d(Vx2 + 1) = V2- dx V77 126 Jo J( •1 X + A/X^ +1 dx (x + A / ^ ^ ~ ; ^ ) V X ^ T T 21 = V2 + I d x = A^ + •id(x + V x ^ +1) ° Dat t = x+ ^f: =i> 21 = V2 + f ^ * " ^ — = V2 + I n d + A ^ ) Ji t 1= Vay f^Vx^Tldx = — + i l n ( l +V2) Jo 2 x^+l 279 T i n h I = X + Vx^ + -2 dx xVx^ + HV Buu chinh Vi&n thong - 1997 Gidi T a CO : Dat I I, = lo = x^+l = -2 xylx^ + -V2 xdx -2 x^ +1 xdx -2 ->/2 -2 dx = -V2 f-V2 -2 Vx^ +1 XA/X" + 1 r - V d ( x +1) -2 x'+l xVx^Tl dx x^ +1 x' +1 dx XA/X^ + -2 X^ +1 = ln I2 = In -2 A/3-1 A/2 1+ , VS-l -In- A/3-1 , I = r ^ - i ^ d x = V§-V^.lni5_l_in J-2 XA/X^ „ J „ + 11 = A/3-V5 '2 dx -V2 Vay dx -2 A/5-1 A^ Jo 127 |28o| T i n h I = •2 x + •0 ^3x dx +2 DH Sa pham Quy Nhan - 1999 Gidi Dat t= ^3x +2 = x+2 X D o i can I = Vay I = Vay ^ X = _ X = X+ ^3x X+ i(t^ at^dt = d x - 2) + = - (t^ + 1) 3P (t^ + Dt^dt J +1 = t = t = +2 E81 3V2 10 15 ^ 42 15 ^/3772 10 x^dx Tinh I = €77 DH Thuang mai - 1997 Giai Dat t D o i can Vay = = x + x^ t = X = 't = V7 x = fV7 x ^ d x _ => 3t^dt = 2xdx x^.xdx 141 f^l^.3t2dt =^ r••^t^-Ddta t 2 J 20 Ji E82 Tinh I = XA/2-x^dx Jo DH khoi B - 2013 128 Gidi Dat t = V2-x^ o Doi can X = X = = - t = t = 1= J^^xV2-x2dx = 283 Tinh I = \ ^ V l + x^dx Vay « tdt = -xdx ^ -t.tdt = V2 rf2 -1 2V2 DH Quoc gia Hd Noi - B/1998 Gidi Dat t = V l + x^ Doi can X = X = Vay I = t^ = + x^ => t d t = 2xdx t = V2 t = f ' x ^ V l + x^dx = f ^ ( t ^ - D t ^ d t = Jo Ji V ] 15 15 J 284 Tinh I = ^ dx ;^xVx2-l BH Bach khoa Hd Noi - 1995 Gidi Dat t = Vx^ -1 ^ x = V2 Doi can Vay = X = ^ t^ = x^ t = 2 t d t = 2xdx t = dx •"vf x V x ^ - l L a i dat t = tanu vdi u e tdt fct^+Dt •1 12' •1 dt dt = (tan^u + l)du t^ + = tan^u + 129 _6_ 08 x - ^ = x/^xc3 - = mi xp ^ + ^ T - I = - = I J t9 = X= T= ^ c=: : OD I + 7/ ^7 = ^ xp = ;p^59 X = c= X X xy^ J x/^x + x/^ J (X-x) -.}) OSI Bx = I uto l o g = * =1^"''!' = x T= ^ = = ^ x/^x + xA-^ - = :>p dx = - d t -fc Vln(3 + t) + V l n O - t) =I = ^ f4 V l n O - xj Suy : I = I , dx = •"2 V l n O - X ) + Vln(3 + x) Gidi Ta CO : tan I= cos '4 XA/I X + cos^ tan :dx = X COS^ X / l + X :dx ^ COS^ X tan u = V2 + tan^ X n Doi can :dx cos^ xV2 + tan^ x Dat X => = + tan^x 2udu = 2tanx —^-— dx cos X "u = Vs X = — n u = Vs X = — Vay = 289 Tinh I = udu >/3 U V5 =u = Vs-Vs V3 'V(r^:77dx £)i? y Hdi Phong - 2000 131 290 Dat X = cost D o i can Vay 1= I = X = X = Gidi dx = - s i n t d t vdi t e 0; t =0 t = ^ f V ( l - x ^ ) ^ d x = [ V d - c o s ^ t)^ s i n t d t = f Vsin^ t s i n t d t f s i n " t d t = - f (1 - cos 2t)2dt Jo JO J It r - C O S t + - (1 + cos t ) d t J 2 cos t + — cos t d t ' ^ — t - sin 2t + — sin t Vay = _ 37t CjoT7fdx=^ 16 Jo x^dx Tinh I = + Vx^ + X HVNgdn hang TP.HCM -2000 Gidi Ta CO : fi I = x^dx X t = Vx^ + => x^(x-Vx^ +l)d> x^-lx^+l) f'x^Vx^ + l d x = - — t^ = x^ + => x^ Vx^ + l.xdx t d t = 2xdx t = V2 X = + Vx^ + '-xMx+ Dat x =1 D o i can t =1 132 Vay I i = f^x^Vx^ + Ixdx = >/2 Vay (t^ - D t t d t = 4V2_2V2_1 x^dx I= rV5 fV2 t^Ct^ - l ) d t _ l , ^ _ V _ l ^ l ^ J _ ^ ^ ^ 15 dx 29ll T i n h I = -J (2x + 3)V4x2 + 12x + DH Bach khoa Ha Noi - 1986 Gidi Dat t = V4x2 +12X + = 4x^ + 12x + => => 2tdt = (8x + 12)dx tdt = 2(2x + 3)dx X = — CO Vay — (2x + 3)^ = 4x^ + 12x + = 4x^ +12x + + = 1^ + : I t = X = Ta t = 2A/3 Doi can dx (2x + 3)dx = 1A (2x + 3)V4x^ + 12x + 2S I = J0 L a i dat tdt t = 2tanu v d i u e D d i can t =0 Vay I= - Nen I= dt Jo {t^+4)t t = 2V3 f2>/3 •' (2x + 3)^ V4x^ + 12x + n n { d t = 2(1 + tan^u)du 2' 2) 71 U = — u = f 2(tan^ u + Ddu _ du = — u _ 7t ^12 4(tan^ u + 1) J0 •\x n_ " i (2x + 3)V4x^ + 12x + 12 133 I I [...]... 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 ^... 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) =... 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 - 19 98 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 ,20 01 31 I Tim ho nguyen ham cua f(x) = (1 + X2 )10 02 DH Quoc gia Ha Ngi - 2000 Gidi 20 01 Ta... - 1 — cos t + — sin t + C 16 4 cost + 1 cos r I 7t^ X + — cos -1 1 — cos 4 3;\ / 16 4 71 ' X + — + I X + — I 1 3^ V3 / + — sin X 4 V + A 3j + c 3; 68 I T i m ho nguyen ham ciia f(x) = ^ sin 2x - 2 sin x BH Hang Hdi - 19 99 Giai Ta CO : « x ) = 2 sin x(cos x - 1) sin x 2 sin^ x(cos x - 1) sinx -sinx 2 (1 - cos^ x)(cos X - 1) 2(cos x - 1) ^ (cos x + 1) A Dat (t-l)2(t + l) it-if B • + C + • t -1 t +1 1 =... + 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/20 01 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. .. 20 01 J f (x)dx = (1^ ^2 )10 02 2 a) Dat x = tana X 10 00 NIOOO X •(l + x^)^ \10 00 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 /19 77 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) ^... Dat u = X+ — , ta x'^ + — + 4 x^ J 1^ dx = X+ — Xy 1 X+ — V \ X+ — X X + — X CO : X f (x)dx = r du u^ + u u ^ + l - u ^ du = fdu udu u(u'' + 1) 2 , I I 1 fd(u' +1) , 1 1 1, , 2 ^ = In I u I -2- J — := In I u I ln(u + 1) + C ^ X + 1 u^ + 1 2 Xy + C = - l n x^ + 2 x ^ + 1 + C 2 2 x^ + 3 x ^ + 1 X+ i +1 u ^ + l X/ 48 I Tim hp nguyen h a m cua f(x) = x' + 3 x ^ + 2 21 Ta CO : f(x)dx = Gidi (x^ + IXx^... [(x^ - 5x + 1) - (x^ - 5x)](x'' - Ddx (x* - l ) d x x(x'' - 5)(x^ - 5x + 1) r (x"* - Ddx (x^ - 5x)(x^ - 5x + 1) 1 f d(x^ - 5x) (x^ - 5x + D 5J 1 fd(x^ - 5 x + D x^ - 5x + 1 x" - 5 x x^ - 5 x = i I n I x*^ - 5x I - i I n I x^- 5x + 11 + C = - I n + C 5 5 5 x^ - 5x + 1 5 1 I T i m ho nguyen h a m cua f(x) = x +X + 1 - D^ (X Gidi Ta CO : X + X + (x-lf x^ + X + B 1 (x-lf x -1 (x-lf 1 = A + B(x + 1) + C(x -... = x^ - 3x + 2 -3 x -1 + 21n| X Ux -1) 2 - l l + Inl X 45 1 T i m ho nguyen h a m cua f(x) = + • + x+2 x -1 1 2 + • ix 2| + C x"^ + x^ DH Dagc Ha Ngi - 19 97 Gidi Ta CO : fix) = (l + x2)-x2 X ^ d + X^) x^+x^ 1 fix) = — ( l+ x2)-x2 X^ Vay X(l+ x^ fdx 1 J X^ 1 x2) 1 X^ •dx fdx f (x)dx = fdx X(l + +• X l + x2) x2 • xdx X2 +1 X d(x2 +1) = J _ - I n l x l + i l n ( x 2 + 1) + x2+l 2x2 2 x +1 46 I T i m hp nguyen... t -1 t +1 1 = A(t + 1) + B(t - l ) ( t + 1) + C(t - 1) ^ 1 = (B + C)t^ + (A - 2C)t + A - B + C A = i 2 B +C=0 A - 2C = 0 B = - i 4 A-B+C=l c=l 4 Vay Hay 1 ( t - i f (t + 1) f(x) = - 2(t-l)2 1 sin 2x - 2 sin x 1 4 ( t - l ) • + 4(t • + l) - sm X sin x + • 4(cos x - 1) ^ 8(cos x - 1) sin X 8(cos x + 1) 29 Nen f (x)dx = dx sin 2x - 2 sin 4(cosx-l) x - — I n I cosx - 1 1 + — I n I cosx + 1 1 + C 8 69 ] T i