Phương pháp tính tích phân và số phức phần 2

292 Tinh I = •7 Jo X sin x X + (x + 1) C O S x , dx S i n X + COS X DHkhoiA -2011 Gidi Ta CO : I = (7 Jo X s i n x + (x + 1) cos x , n dx = X s m X + cos X Jo Ii= pdx = x Jo X X cos X s i n X + cos X Jo s i n X + cos X cos x dx X ~ s m X + cos - +1 du -dx Dat u = xsinx -1- cosx => du = xcosxdx X = In u = ln V2 (7 X s i n X - - ( x - f 1) cos X , 1= Tinh I = X + X s i n x + cos -dx X _ 71 xcosx X u = l x =0 D o i can 71 X = — u = V2 I2 = Vay 93 •* Jo X sin X + cos x rr 4- X s i n x , dx 7t V2 f 7: v4 y - In = In , V2 ^4-1 dx = — + I n — 4 cos^ x Jo BHkhoiB -2011 Gidi 7C Ta I CO : Ii = h = \ X sin x cos^ x — = tan ^ cos * 71 dx dx = * cos^ sin X dx cos^ X X * X = V5 X X xsmx cos X u = X => du = dx , dv = sinx , , — dx chpn v = cos sin x dx = cos^ X cosx X 134 - X Sin X, Jo C O S X dx = 294J I = Tinh I = Jo cos X X Jo , s i n x - l 2K Vay COS _ 271 dx x sinx + n d(sin x) Jo s i r sill X - d(sin x) = 271 X- sin X-1 sin In J ln(2 - VS) - •0 + X s i n Xdx cos^ X = V3+ — + ln(2-V3) dx - ^ l + x + x^ + Vx* + x ^ + Gidi Dat x = - t => dx = - d t Vay = Suy r a : 21 = Ta x= - l t = X =1 t = -1 dx -dt ^1 ^ Doi can 1- t + -1 + Vt" + St^ + •'-^ + X + x^ + Vx^ + x ^ + [g(x) + g ( - x ) ] d x vdti g(x) = 1 + X + X^ 3x' +1 CO : g(x) + g ( - x ) = + X + x^ + Vx"* + x ^ + 1 - X + x^ (1 + X + x^ + Vx'* + 3x2 + X + x^ + Vx'' + Vx" + 3x^ + + ^ - X + x^ + A/X^+3X^+1 + 3x^ + _ X + x^ + Vx^ + Sx^ + 1) 2(1 H-x^ + Vx^ +3x^+1) (l + x + V x ^ + x + l ) - x 2(1 + x^ + V x ^ T s x ^ T l ) + x" + x" + 3x2 + + 2x2 + 2Vx^ + x + + 2x2 Vx^ + x + - x2 2(1 + x2 + AM+3X2TI) + 2x* + 4x2 ^ 2Vx*' +3X2+1 + 2x2 x ^ ^ x + I + x2 + Vx^ + 3x2 ^ + x^ + 2x2 + Vx^ + x + + x2Vx* + x + 135 1+ + Vx" + 3x2 ^ J (i + x^+ V77377T) + (x2+ xnx2V77377T) Do : Dat 21 = J-i + •1 dx •'0 + ^ X = t a n u => dx = (1 + tan^u)du + tan^ u I = i + x' dx Jo l + X^ fx = D o i can X = n Jo 295I T i n h I = u = U = 71 — 71 Jo dx ^ x V x^ + fiHAn ninh -1999 Gidi Bat t = Vx^ + => X = x = >/7 D o i can = x^ + 1= Vay 1= t-3 = i l n t + Vay Tinh I = 2tdt = 2xdx t = ^ t = dx 4 (t^ - 9)t tdt Esiel " e l dx 7J dt = - In - ^ x y f ^ ^ dx rl + X + V l + x^ HV Khoa hoc Qudn sU - 1998 Gidi Ta (1 + x) - V l + x^ (1 + xf d + x dx CO : • ' " ^ l + x + V l + x^ + x - V l + x^ fl > -1 u ^ (1 rl 2x -1 dx pi - +1 dx- -dx -1 2x^ 136 = - - + dx dx = - ( l n | x | + x) J 2J-1U J-i -1 X = t d t = 2xdx t=^ x =1 t = Vay I , = r i i ^ „d—xax= = J-i 97 1= I ^ f f t-^4d t^ = J - ' ^^ 22 (( tt 22 - l ) 2x2 r — ^ - • ' - ^ l + x t V l + x^ -I Tinh I -1 2x2 e= Nen I f i x V l + x^dx Tinh l = Doi can = f d x B i e t fl(x) = DHXdydung - 1999 Gidi ft Ta CO : fllx) A = V x ^ Vi x e V3 Vs nen x > T xdx Dat t = V l - x ^ X = X = Doi can ^ V8 ] Si -tdt _ [-3 " J l ( - t )t = t = 2tdt = -2xdx l t = i fg dt 1, t - t + ^ = i l n ^ 1 2 137 §9^ Vay V8 _ J— f V3 ^ dx = - I n — 2 T i n h = j " ^e'^ - I d x Gidi D a t t = V e " - => t^ = e" - V i t^ + = e" => dx = D o i can Vay X t d t = eMx 2tdt t^+l = In X = t = Ve" - = t = Ve" - = ffin in 2 I f•1l tt^^d tt Ve" - Idx = f = 1=1 Jo Jot^+i r l | (d t - Jo dt t^ + D o i b i e n dat t = t a n u => d t = (tan^u + l ) d u D o i can fl t = t =0 n u =— u =0 • - ( t a n ^ u + 1) du = dt ot^+l '7 , tan^ u + I= fin I Ve" - Idx = •4du Tinh I = f =4 Vay [299I = 2-2.^ = - ^ dv ^ (2" - ) V - ^ - " Gidi X Ta CO : 22 I = :dx = * (2" - 9)73-2^"" Dat t = V3.2" - dx * (2" - 9)73.2" - =^ t ' = " - => 2" - = 2tdt => ^ ^ = 2"dx 31n2 t^ - 138 Doi can Vay = X = t = X = t = 2tdt f2 t 51n2 300 Tinh I = •In In t-5 — In t + I n 10 dt In J i t ^ - S ^ _9_ 14 2" - 2" r2 r2 4"" -dx -2 Giai Dat du — = (2''In u = 2" + 2-" D o i can Vay 1= X = X = 2-'')dx va 4" + 4"" - = (2" + 2"'f u =— 17 u = — 2" - 2" •2 - •dx = ^ Ji 4" + 4"-" - du _ - ' In 2 du In Ji - 17 , 81 •In— hi 25 u-2 •In In u + Tinh I = 2e^''+e^''-l In -dx + e^" - e" + JO Giai Ta CO : Vay 2e3- + e^^ - + e^" In I = J- 26^" (e^'^ +1 ~ - e ' +1)' +6^'' + Jo In - X In Tinh I 2x^ - 3x^ - X -1 •ln2d(e^=' + " - e " + ^ " - ! = ln(e="= + e^" - e'' + 1) 302 +1 e^" + e^'' - „3x e , „2x +e „x - e +1) , 1 rln2 JO dx , 1 =^"T- + Idx 139 Dat Ta Gidi t = - X =>dt = -dx CO : 2x^ - Sx^ - x + = ( - tf - ( - t ) ^ - ( - t ) + = - t ^ + at^ + t - Doi can Vay 1= t = t = x = X = £ ^ x ^ - Sx^ - x + I d x = ^ ^ - ( t ^ - St^ - t + l ) ( - d t ) Jo ^2t^ - t ^ - t + I d t = - I S u y r a : 21 = => I = I = soil T i n h I = Jo dx fO ^ t ^ - St^ - t + I d t = "^^ + V - x ( l + x ) Gidi Ta CO : I = dx fO + V - x ( l + x) "'-^ + dx n - Dat ^ + X — + X = - s i n u => d x = - c o s u d u 2 Doi can X = 71 U = — u = — x = Vay 1= t — cosuau F ^ Jo 1 , + J sin'^ u V4 p - ^ ^ d u = Jo + c o s u J Jo - + cos u du ^-2p ^" Jo + cos u Tinh J = Jo + cos u u = Dat => d u = '^^^ 1+ e t = tan— Doi can t = 71 U = — t = 140 2dt + L a i dat 71 t = a =— [I f t a n ^ A/3 Jo a +1 tan a +1 da = dx I 2x + •1 18 7i(9 - 2A/3) 18 + x) V-x(l 304| T i n h I = a +l)da "a = rt = o J = D o : => Sitan^ t = A/3 t a n a Doi c a n Vay + t^ dx V2x - x^ Gidi Ta T I = CO 2x + , dx = •'iV2x-x2 •1 2x - + + V2x - x^ fi rl 2x-2 >/2x - x^ * Tinh I i = 2x-2 •1 dx + dx dx 2x-x^ dx V x - x^ Dat u : V2x - x^ => u = 2x - x^ Doi c a n X = Vay Tinhl2 = ^ •1 -2udu Ii * u = X = — •1 u => 2udu = (2 - 2x)dx A/3 u = = - 73 du = - u dx = -2 1- dx = (x^ - 2x + 1) Dat x - = sint dx = costdt Doi can = A/3-2 dx ^ Vl - (X - x =— X = 1)^ " t = 141 305| Vay « "i Do : I = [-1 cos t d t Vl - = - ^ dt = 5t sin^ t 2x + dx = V - + ^ i >/2x Tinh I = ^ A/-3X^ + 6X + Idx Gidi T a CO : f V-Sx^ Jo Dat + 6x + I d x = ( ^j4-3(x-lfd\ f , - - (x - l)^dx Jo Jo V Vs — ( x - 1) = s i n t => dx = - = c o s t d t V3 D o i can Vay X = X = t = 1=2 sm 2~ t - = cos t d t = rO V5 J cos^ t d t fO l + cos2t (1 + cos 2t)dt tdt = • ^ V3 J V3 J (n ( t + - sin 2t — + V5 306| T i n h I = ''Vx^ - x ^ S — +xdx DHThuy M ~ A/2000 Gidi V x ( x - D^dx = f ^ V x { l - x ) d x + I = X2dx- = i x^dx + rl Jo - - x - x o + 0 fVxCx-Dd i x^dx - C2dx -2j o -X2 = 142 Gidi Dat t = V x - X = t = X = t = +1 •it^_+_t Doi can I = 308 fU^ = X - 2tdt = t + TInh I = x - - t + 2tdt = dx • o dt = - t + - ^ — dt = — - 4ln2 t + l j Inxdx, DH Khoi D -2010 Gidi •e ( T a CO : I = h = 309 «lnx •Ji X I = Tinh I = Jo f •1 •e h Inx xlnxdx dx dv = xdx r J xdx = Ji 2 +1 v dx D o i can u = X = e u = f -A 3^ x - - I n x d x = e"^ + if x = 1 udu = (x + V = dx •3 + I n x dx -Inx u = I n x =:> d u = h = Vay Inxdx = u = I n x => d u = Ii = Dat 3^ xlnxdx Ii = Dat x - - -3.- = -2 dx DH Khoi B - 2009 143 |06| Cho In = j ^ ^ x ^ d - x ^ f d x ; Jn = £ x(l - x ^ f d x T i n h J n va chijfng m i n h bat dang thufc !„ < T i n h !„ + i theo !„ va t i m l i m 2(n +1) n = 0, 1, 2, vdi V n = 0, 1, DH Can Tha - 2000 Gidi T a c o : T a CO : J ^ = - [ ' ( - x ^ ) " d ( l - x^) = - Jo 0 V = rl Vay 2xn n + 2{n + l ) In ~ +l 2(n +1) - X x(l-x2)"x^dx = Jo (1) In + - 2(n +1) d-x") 2(n + 1) J In + + ~ x^d-x^f^Mx H ^n2n + I„^.i 2n + , l i m - ^ ^ = lim = x->x x^=o2n + Cho In = f^x^e-^Mx Jo n = 1, 2, 3, a) C h i i n g m i n h In > In + i T i n h In + i theo In 252 b ) Chufng m i n h < !„ < v d i Vx > Suy r a l i m I „ (n - De^ n->a3 DH Quoc gia TP.HCM - 1997 Gidi a) V i < X < < x" " ' < x" x" * 'e-'" ^ < x"e-''' f^X^^^e-^^dx < f^X^e-^Mx Jo Tinh Dat I„ ^ theo I„ u = x" * ^ ^ Vay (vi 6-2" > Vx) Jo T a c6 : In^l lim n —km In = v d i n l a so' nguyen > a) T i m he thiJc l i e n he giuTa In + i va ! „ b) Tinh liml„ BH Tong hap TPHCM - 1991 253 Dat b) Ta a) CO : !„ + i + rl = Gidi , - ( n + l)x 1+ e • dx + 2x rl 1+ e 2x lg-2(n.l)x^^g2x) ) CO k = - ( n + 1) nen dx ,-2(n + llx dx = + e^" ~1 -1 g - ( n + l)x 2(n + 1) Ta 2(n + 1) (e (e-2'"^>'-1)= n->K2(n + l ) lim dx -2(n + l) -1) k h i n - > oc t h i k - > - x -1 : l i m (!„ + i + I n ) = l i m Ma = l i m In lim i-(e'-=o Dodo loOOJ C h o a, b l a h a n g so Chufng m i n h limln^O lim e"^ s i n n x d x = ( v d i a < b ) Gidi I„ = Vay u = e''^=> d u = x " ' d x ; Dat e" a * Ta CO : IH J = dv = sinnxdx v2 s i n n x d x = - — e"' cos n x n — I x2 e — - —(e'' n cos n x e n b2 1 |cos nb| + - e b +— fb V = — cosnx n x.e" cos n x d x n cos n b - e^^ cos n a ) •\ a2 ma |cos na| J cosna < va Icosnb | < =^ Ma * < I H J < -(e'^' + e " ' ) n l i m - (e^^ + e^^ = Vay l i m Hp = (1) M a t k h a c f ( x ) = x e " cosnx l i e n t u c t r e n [a; b ] V a y l u o n t o n t a i m , M € R c h o m < x e " c o s n x < M => fb m ( b - a) < m ( b - a) xe" ^ < n cos n x d x < M ( b - a) •b J „ = n - J xe ^2 2M(b-a) cos n x d x < n a 254 , 2m(b-a) , M ( b - a) ^ Ma h m = lim = T i r ( l ) , (2) suy r a I l i m I„ = l i m b ^ Suy r a ,• T n h m Jn = (2) e" sin nxdx = f k+ 1 510| a) Chiing m i n h V k > thi Ink < I n xdx < l n ( k + 1) b) Churng m i n h : (n - 1) ! < n " e ' " " < n ! Suy r a co lni- >m Vn e N n DH Tdi chinh TP.HCM - 1992 Gidi a) Do y = Inx l a h a m so' dong bien t r e n (0; +=o) nen V k > : k < x < k + t h i Do [(k + 1) - k j l n k < f k+i b) Do k e t qua cau (a), t a c6 I n k < I n x < I n (k + 1) I n xdx < [(k + 1) - k ] l n ( k + 1) Inl< J I n xdx < l n (1) ln2 < I n xdx < l n (2) J2 l n ( n - 1) < n-l I n xdx < I n n Cong ve theo ve (n - 1) b a t dang thufc t r e n t a duoc I n l + ln2 + + l n ( n - 1) < I n xdx + < ln2 + ln3 + o Ma ln[1.2.3 (n - 1)] In = < I„ = f3 I n xdx + + Jn-l I n xdx + Inn I n xdx < ln(2.3 n ) I n xdx = x ( l n x - 1)1 ° = n ( l n n - ) + ' ^ Do l n [ ( n - 1)!] < n l n n - n + < l n ( n ! ) ^ e ' " " " - ' ' " < e " ' " " - " * i n-e^-" > n ! (n - 1)! < e ' " ' " " ' e ' ' " < n ! => n = n! (n-D! n 1! n! = > — n"e^"" n! 255 Isn] Taco: n , e° l i m ( e ° e"^) = — = e n-»co M a t k h a c x e t h a m so e y = x'' vdi x > (1) T a c6 : I n y = - I n x X Suy r a l i m [In y] = l i m = lim lim [Iny] = lim — = = I n l n->oo Ma n->« X lim[Iny] = lim [Iny] = I n l n—>x D o Nen n->cc l i m y = l i m x" n-*oo - - l i m n " e"e = n-»co ^ = lim n" lim e ^ = - n->oo n-*K n-»te TCr ( ) , ( ) n g u y e n l i k e p t a c6 lim n->=o Chutng m i n h r a n g , V n € N t h i : !„ = DH Y dicgc TP.HCM Q yfn^ n (2) = - e '"\2x-l)2""ie''-'''dx 1981 = + DH Thai Nguyen - 1999 Gidi D u n g p h i r o n g p h a p chufng m i n h q u i n a p * K h i n = t a CO I i = Dat u = (2x ( x - l ) e ' ' - ' ' dx - ir du = 4(2x - l ) d x dv = ( x - l ) e ' ' - ' ' ' d x Vay Gia SIJT - 8° • ( x - De^-^^dx + Ii = -(2x-ire I i = -e° + * e"-" = _4(e0 - e°) = dCx-x^) = - " I k = v - d t = dx => X = - t I : f'x(l-x)^^dx= Jo I = rl (t^^-t20)dt = D o i can t = x = t = X = f°(l-t)ti9(-dt) Ji /^20 ^21 20 21 420 (1) 259 b) Theo n h i thiJc N e w t o n t a c6 : x ( l - x ) ' ^ = x(C°9 - C i g X + 0^9x2 - Cl«x^«-C-xi^) pl8 x ( l - x)^^dx = 19 19 X -c 19 p 19^20 21 \ 21 p 19 Tir(lU2)tac6: i C SITI Cho I„ f - i C x^d + x^fdx, Chu-ng m i n h r a n g : + - C 21 (2) =-i^^ 420 n>2 - C ° + - C | , + - C ^ + + -C" = 3n + " 3(n + l ) BH Ma Hd Noi - 1999 Gidi D a t 2, du = Sx'^dx u = + x^ D o i can u = u =2 x == X == du 1 '^u"du ~ 3' n + •1 = x2dx 3(n + 1) n + 2""^ - 2""^ - (1) Theo n h i thufc N e w t o n t a c6 a ^ + x^r= c ° + c j , x + c^x'^+ + c>3" x2(l + x3)"= C ° x + C 1x5+C^x^+ + C ^ ^ " ^ ,3n + .+cr, 3n + c; •x2(l + x3)"dx = C ° = icO+icl+ic^+ + Tir(l), (2)tac6 : ^ C ^ + " J, + " " +3 3n + +—i—C = 3n + (2) 3(n + 1) 260 p i E N TICH HINH PHANG, THE TiCH V A T THE TRON XOAY A C O N G THL/C TfNH DIEN TfCH HINH PHANG PhUcfng phap ; Cho ham y = f(x) (Ci) y = g(x) (C2) lien tuc tren [a; b] thi dien tich hinh phang gidi han bdi (Ci), (C2) va hai dudng thfing x = a, x = b la : S = fV(x)-g(x)|dx Ja Ghi chu : a) De bo dau tri tuyet do'i ham so dudi dau tich phan, thi ta phai xet dau f(x) - g(x) tren [a; b] hoac nhd thi ta thay diTdng (Ci) (i = 1, 2) nao nkm tren b) Neu de bai khong cho dudng th^ng x = a, x = b ta phai tim giao diem ciia (Ci), (C2) trUdc tien C, : y = f(x) c) Khi phiTcfng trinh f(x) - g(x) = v6 nghiem tren (a; b) thi S = f(x)-g(x)| dx = [f (x) - g(x)]dx d) Khi phUcfng trinh f(x) - g(x) = cd nghiem x = c (c e (a; b)) thi [f (X) - g(x)ldx S = [f(x)-g(x)]dx e) Ta phai ve thi de thS'y [f(x) - g(x)] duong hay am cac ham so cd tri tuyet do'i Sisl Tinh dien tich hinh phang gidi han bdi dudng cong cd phiTcfng trinh y = sin^xcos^x, true Ox va hai dudng thSng x = 0, x = - DH Bach khoa Ha N6i - 2000 261 Vi < X < Vay S = => Gidi sinx > 0, cosx > sin^ x cos^ xdx = => sin^x.cos^x > sin^ X cos^ xd(sin x) sin^ x ( l - sin^ x)d(sin x) = - sin^ x — sin X — (dvdt) 15 |519| T i n h dien t i c h h i n h p h ^ n g gidi h a n bdi y = (e + l ) x va y = (1 + e'')x DHKhoiA -2007 Gidi PhUcfng t r i n h hoanh giao d i e m : (e + l ) x = (1 + e'')x c:> xCe" - e ) = x = 0, x = l T a t h a y k h i < x < t h i (e + l ) x > ( + e'')e'' (do => ( + e > (1 + e") Do : S = 0e'' (1 + e)x > ( + e'')x) f [(e + l ) x - (e'' + l ) x ] d x = e f x - f^xeMx = Jo Jo Jo (dvdt) [52oj T i n h d i e n t i c h h i n h phSng gidi h a n bdi y = x^ - 2x va y = -x^ + 4x DH Mo Dia chat - 1997 Gidi Phuang t r i n h hoanh (io giao d i e m : x^ - 2x = -x^ + 4x Ta CO : x 2x2 _ 2x^-6x = - t h a n h hai phan T i n h dien tich h i n h phang cua moi h i n h DH Kinh te Qudc dan Ha Ngi - A/2000 Gidi Phirang t r i n h h o a n h giao d i e m cua parabol va dudng t r o n 1^ : x^ + 2x - = X = o X = - (loai) Vay parabol c^t ducrng t r o n t a i h a i d i e m A(2; 2), A'(2; - ) Dudng t r o n cAt true h o a n h t a i h a i d i e m N ' (2V2; ) , N {-2^|2• 0) Ca diTcJng t r o n va parabol deu n h a n true hoanh l a m true do'i xufng D i e n t i c h cua p h a n h i n h phSng O A N A O dugc t i n h b a n g cong thiifc : Si = Ta CO I = ! , V2x.dx+ c2j2 ^f2x.dx = X A / X I 7- V8-x'^.dx ^ 265 Ta CO K = •272 Dat X = 2V2 sin t Doi can => dx dx = 2^2 cos t x = 2V2 x =2 -a/2 Vay : 71 K = } cos^ t.dt = Vay A 2(1 +cos 2t)dt = t + - sin 2t Si = 2(1 +K ) = - + 71 - — V3 ^ = 71-2 - - (dvdt) 3J Dien tich phan lai ciia hinh tron ngoai parabol la : \ ( S2 = 7:(2>/2)^ S,Si = 67t -< ^ + 3; (dvdt) |528i Tinh dien tich hinh ph^ng gidi han bdi cac dudng y = | x^ y = mp Oxy 4x + i va BH Su pham Hd Ngi - B/2000 Gidi PhiTcfng t r i n h hoanh giao diem : |x^-4x + 3|=3 Vay S = x = 0, x : = o (3-|x^ - x + 3|)dx Do tinh do'i xiJng qua dudng thang x = nen r2 ^ (-x^ + 4x)dx + f ^x^ - 4x + 6)dx = '5 S = ' ( S - l x ^ - x + 3|)dx S = r = (dvdt) 529I Tinh dien tich hinh ph^ng gidi han bdi true tung, y = 2" va y = - x HV Buu chink Viin thong - 1999 266 [...]... = Dat , ; 2v V l + sin xdx = 1= ^ X X N2 (^X ^\ 71 = 2sin2 — + — l2 4 J sin — + cos — 2 2J Vay _ = sin — + cos — + 2 s i n —cos — 2 2 2 2 2sin^ X 71 V2^4y dx fx 71 ^ sin — + — dx ^2 4J p2jt 0 71 t = - + 2 4 => dt = dx Doi can 2 X = 5jt Vay I = 2V2 I = - / _ 0 t = t = ^ 2 Ii 4 571 |sin t| dt = 2^ 2V2 C O S t I + — 2V2 cos t 4 I349I T i n h I = X = 27 1 2^ 2 , sin tdt - ^ sin tdt 5it 4 = 7t 2^ cos xVcos... : I = t r 2 0 Jt "2 0 (1 + cos 2x 1 - cos 2x - — s i n 2x 2 1 2 X dx 1 cos 2x — s i n 2x dx 2 2 3 3 1 — s i n 2x + 2 2 '71 ^ cos 2x1 ^ J o ^ 2 2 V 2 167 358 Tinh I = 2 sin^ X CDs'* xdx , DH Ngoai nga Ha N6i - 1996 Gidi Ta CO : sin^xcos''x = -8 (1 - cos 2x)(l + cos 2xr = - (1 - cos 2x)(l + 2 cos 2x + cos^ 2x) 8 = -(l-cos2x) 1 + cos 2x + 1 + cos4x 8 V4y I = - 359I Tinh I = 1 r 1 + -1 cos 2x - cos 4x... cos Jt - du = 6sin2xdx Doi can Vay 0 x sin2x Tinh I = ~ 3 - sin 2x + sin x , 2 dx Jo Vl + 3 cos X BH Khoi A - 20 05 Gidi Dat t = Vl + 3 c o s x =i> 7t X = — t =1 Doi can X = 2 0 = 1 + 3cosx => 2 t d t = -... DH Quoc gia TP.HCM -20 00 Gidi Ta CO :I = 4 2( l-cos2x)Mx= 0 '3 = I 8 355 4 J — 2 2 cos 2x + — cos 4x dx 4 1 1 ^ _37r — sin 2x + — sin 4x 4 32 / l 6 2 X 2 cos^ 2xdx, Tinh I = DH Kinh te TP.HCM - 1993 Gidi Ta CO : I = 2 2 (1 + COS 4x)dx = - ( 2 0 X + sin4x'| 2 _ 1 - + — sm 27 1 4 ) ~ 2 U 4 71 4 166 356| Cho I = 2 cos^ X cos^ 2 x d x ; J = a) T i n h I = I + J , I - J 2 sin^ X cos^ 2xdx b) T i n h I va... ( 2 (cos^ x + sin^ x) cos^ 2xdx = p cos^ 2xdx Jo Jo fr 1 + cos 4x , 2 dx = Ta CO : I - J = 2 (cos 1 - U X + sin4x^ 2 _ n 8 X - s i n x) cos^ 2xdx = s i n 27 r _ 8 71 ~4 2 cos 2x cos^ 2xdx 0 D d i b i e n , d a t u = sin2x X = D d i can 71 — => du = 2cos2xdx u = 0 2 u = 0 x = 0 2 (1 - sin^ 2x) cos 2xdx = - f (1 - u^)du = 0 2 Jo V4y 0 I - J = T 357 T I +J = 4 I - J = 0 b) T a c6 : Tinh I = 2 8 8 (2. .. g(x)] dx - J^^ [ f (x) - g(x)] dx -1 I = 34ll Tinh I = r (x^ - 2x2 - x" 2x^ x^ 4 3 2 X + 2) dx ' , + 2x f2 (x^ - 'x' 4 -1 2x2 - X + 2x2 ^2 3 2 2)dx + 2x 37 12 VT - sin x d x DH Ngoai thuang - 1994 Giai Ta CO : I = sm COS — 2) dx = X sin x dx COS — 2 2 159 ^p^SOD X p X UTS 2 — = K = UBD log XSOO : 0 f =1 = x p |X U T S | : OD -J •J U T S 2/ ^-3- J TpG x 1^8 nis I u SOD - ) + 6 - ;pi SOO BX ^aisgyNg =... cos 6x^ 2 2 16 1 + - COS 2x - cos 4x — cos 6x dx 2 2 — x + — sin 2x sin 4x sm 6x 161 4 4 12 cos^ x cos 4xdx 2 71 32 DH Ngoai nga - 1998 Gidi 1 1 I = - 2 (1 + cog 2x) cos 4xdx = - M (cos 4x + cos 4x cos 2x)dx 1 (cos 6x + cos2 2x) 21 JoT T T cos 4x + — Jo dx 2 (1— sin 4x + — 1 1 ^ 24 sin 6x +4— sin 2x = 0 U ^eol Tinh I = f 2 (cos^° x + sin^° x - cos* x sin* x)dx DH Sa pham Hd Ngi - 20 00 168 Gidi 2 (cos^°