200 bài tập tích phân môn toán 12 ôn thi THPT

200 bài tập tích phân môn toán 12 ôn thi tốt nghiệp THPT tham khảo

www.MATHVN.com Bi Nguyờn hm - Tớch phõn cú li gii TP1: TCH PHN HM S HU T Dng 1: Tỏch phõn thc Cõu x2 I = x x + 12 I = + 16 dx = ( x + 16 ln x ln x ) = + 25ln 16 ln x x Cõu I = Ta cú: dx x + x3 I = I = 1 x + + x x3 x2 + 3 + ln( x + 1) = ln + ln + 2 2x 1 3x + x x 5x + Cõu = x ( x + 1) I = ln x Cõu dx dx 13 14 I = ln + ln + ln 3 15 xdx ( x + 1)3 x x + 11 1 Ta cú: = = ( x + 1)2 ( x + 1)3 I = ( x + 1)2 ( x + 1)3 dx = 3 ( x + 1) ( x + 1) Dng 2: i bin s Cõu I = Cõu I = ( x 1)2 (2 x + 1)4 dx ( x 1)99 101 ( x + 1) 7x I = 2x + 99 x Ta cú: f ( x ) = 2x + I = Cõu I = (x 5x 2 + 4) x7 (1 + x )5 99 7x 1 7x = d ( x + 1)2 x + x + dx 100 Cõu x x I = +C 2x + 2x + dx 1 7x = 100 x + 100 = 900 dx t t = x + I = dx t t = + x dt = xdx I = Bieõn soaùn: Thay Tran Sú Tuứng - Trang 1 (t 1)3 1 dt = t5 25 Bi Nguyờn hm - Tớch phõn cú li gii www.mathvn.com Cõu I = x (1 x )6 dx t t = x dt = 3x 2dx dx = Cõu 10 I = 2 x7 Cõu 12 I = x (1 + x ) 11 t t8 (1 ) = t t dt = 30 168 t t t + dt = ln 32 dt I = 10 t t = x I = 2 t (t + 1)2 x ( x + 1) x.( x10 + 1)2 Cõu 13 I = dx Cõu 11 I = 3x I = t t = x I = dx x ( x + 1) dt x dx 128 t I = dx t t = x I = dt 7 t (1 + t ) x (1 + x ) dx (1 x ).x dx x (1 + x ) t : x = I = t 3 t6 dt = t2 + 1 117 41 + t t +1 dt = 135 12 t + x 2001 Cõu 14 I = (1 + x )1002 x 2004 I = dx 1002 x (1 + x ) Cỏch 2: Ta cú: I = dx = 1 1002 x + x 1000 + x2 11+ x Ta cú: 1+ x 1+ x4 x2 + dt = x3 dx 11 x 2000 xdx t t = + x dt = xdx 2000 2 (1 + x ) (1 + x ) (t 1)1000 I = 1000 dt = 21 t t t Cõu 15 I = dx t t = 1 d = t 2002.21001 dx 1+ = x t t = x dt = + dx x x x + x 3 t I= = dt = ln = ln 2 2 1t t + t + 2 + t dt 1 Bieõn soaùn: Thay Tran Sú Tuứng - Trang www.MATHVN.com x2 Cõu 16 I = 11+ x4 x Bi Nguyờn hm - Tớch phõn cú li gii dx 1 dt = x t t = x + dt = dx I = 2 x 1+ x x t + x + x2 du 5 t t = tan u dt = ; tan u = u1 = arctan 2; tan u = u2 = arctan 2 cos u Ta cú: u2 I= 2 Cõu 17 I = u1 x Cõu 18 I = x4 + Ta cú: x6 + 1 dx dx x6 + x4 + 1 x Ta cú: I = dx t t = x + I = ln x +x x 1x+x 2 (u2 u1 ) = arctan arctan 2 du = = ( x x + 1) + x x6 + = x4 x2 + ( x + 1)( x x + 1) + x2 x6 + = x2 + + x2 x6 + 1 d (x3 ) I = dx + dx = + = (x ) + 4 x +1 Cõu 19 3 I= x2 x4 I= 3 x ( x 1)( x + 1) xdx x + x +1 1+ Ta cú: 0t dx = x2 + x +1 x x +1 1 + dx = ln(2 3) + 12 x x +1 1 dt 11 = t + t + dt t + + dx 1+ = 3 t t = x I = x x2 + 1 I = 2 Cõu 20 I = Cõu 21 I = dx x2 + x2 x2 t t = x 1 dt = + dx x x2 dt +1 t t = tan u dt = du cos u I = du = Bieõn soaùn: Thay Tran Sú Tuứng - Trang = Bi Nguyờn hm - Tớch phõn cú li gii www.mathvn.com TP2: TCH PHN HM S Vễ T Dng 1: i bin s dng x Cõu 22 I = dx 3x + x x I = dx = x (3x x 1)dx = 3x 2dx x x 1dx 3x + x + I1 = 3x dx = x + C1 1 + I = x x 1dx = x d (9 x 1) = (9 x 1) + C2 18 27 I= (9 x 1) + x + C 27 x2 + x Cõu 23 I = x + x 1+ x x x2 dx = x2 + I1 = 1+ x x dx 1+ x x 1+ x x x 1+ x x Vy: I = ( dx = 1+ x x 2x + Cõu 24 I = 01+ 2x + 2x ) + + 4x + 01+ x ) + x x + C1 3 t2 + t dt =2 + ln 12 t t = x + I = ln t: t = x I = ( t t ) dt = 1+ x 1+ x x t t = x + I = dx Cõu 27 I = ( +C Cõu 26 I = x x dx dx d (1 + x x ) = + x x + C2 3 1+ x x dx Cõu 25 I = 1+ x x dx t t= + x x t = x x x = (t 1)2 x 2dx = 4 4 (t 1)dt = t t + C = + I2 = x dx + 15 dx t +t 11 t t = x dx = 2t.dt I = dt = t t + ln dt = t +1 1+ t 0 Cõu 28 I = x dx x + + x + Bieõn soaùn: Thay Tran Sú Tuứng - Trang 4 t(t 1)dt www.MATHVN.com Bi Nguyờn hm - Tớch phõn cú li gii t t = x + 2tdu = dx I = 2t 8t 1t x Cõu 29 I = + 3t + 2 dt = + ln t +1 dt = (2t 6)dt + x + 1dx 1 t7 t4 t t = x + t = x + dx = 3t dt I = 3(t 1)dt = = 28 0 3 x2 + 1 x 3x + Cõu 30 I = dx t2 +1 2tdt 2tdt t t = 3x + dx = I = 3 t t 4 21 t 100 = t t + ln = + ln 93 t + 27 Cõu 31 I = 2x2 + x x +1 = 24 dt ( t 1) dt + 92 t dx x + = t x = t dx = 2tdt t 2(t 1)2 + (t 1) I = 2tdt t 1 2 4t 54 = (2t 3t )dt = 2t = 5 x 2dx Cõu 32 I = ( x + 1) x +1 t t = x + t = x + 2tdt = dx I = (t 1)2 t3 Cõu 33 I = 2tdt =2 x +1 + 2x ) (1 + 2 t3 1 16 11 t dt = 2t = t t dx t 2t t t = + + x dt = dx = (t 1)dt v x = + 2x dx Ta cú: I = (t 2t + 2)(t 1) t 3t + 4t 4 dt = dt = t + dt 22 22 2 t t2 t2 t2 = Cõu 34 I = t2 3t + ln t + = ln t x x +1 dx Bieõn soaùn: Thay Tran Sú Tuứng - Trang Bi Nguyờn hm - Tớch phõn cú li gii www.mathvn.com ( ) x I= dx = x + ln x + x + x2 + x +1 = + ln ( + ) ln ( + 3) Cõu 35 I = ( x 1)3 x x dx I = ( x 1) Cõu 36 I = x x dx = ( x x + 1) x x ( x 1)dx t t = x x I = x3 3x + x x2 x + 1 2 15 dx ( x x )(2 x 1) dx t t = x x + I = (t 1)dt = x2 x + 1 I = Cõu 37 I = x 3dx + x2 t t = + x x = t xdx = 3t 2dt I = 38 (t 4t )dt = + 23 25 Cõu 38 I = dx x + + x2 1 + 1 + x + x2 11 + x2 Ta cú: I = dx = dx = + dx dx 2 2x x 2x (1 + x ) (1 + x ) 1 + I1 = + I2 = 1 + x + x2 11 1 + dx = ln x + x |1 = x 1 + x2 dx t t = + x t = + x 2tdt = xdx I2= 2x Vy: I = t 2dt 2 2(t 1) =0 Cỏch 2: t t = x + x + Cõu 39 I = Cõu 40 I = 1 3 x (x ) x4 x2 dx x2 dx x Ta cú: I = I= 1 Ta cú: I = dx t t = I = x x x t(tdt ) t2 x xdx t t = t2 x t = x tdt = xdx t2 = dt = (1 + )dt = t + ln 2 t+2 t t 3 = + ln + 3 Bieõn soaùn: Thay Tran Sú Tuứng - Trang www.MATHVN.com Cõu 41 I = x ( x + 1) x + Cõu 42 I = Bi Nguyờn hm - Tớch phõn cú li gii 27 x x+ x x2 + x + dx t t = x + x + x + I = 1+ 1+ 2dt = ln(2t + 1) 2t + x2 + x )2 (2 + + x )2 (1 + 4 = ln 3+ 3 dx 42 36 t + + x = t I = 2t 16 + dt = 12 + 42 ln t t 3 x2 Cõu 45 I = 2( x + 1) + x + + x x + 2t (t 1)2 dt t t = x + I = t(t + 1)2 Cõu 46 I = 2 x x + 2011x x4 Ta cú: I = 2 M= 2 2 2011 x3 I= x x3 N= 1 x2 x3 dx + 2 dx = 2 = (t 1)2 dt = (t 1)3 = 3 2011 x3 1 2 dx dx t t = dx x2 dx = M + N M = t 3dt = 2 2011 2011x dx = x2 3 = 213 128 14077 16 14077 21 16 128 dx Cõu 47 I = (1 + x ) + x 3 3 = 15 ln 2 2t dt = + dt = + ln 12 t t2 + t2 + t(t + 1) Cõu 44 I = t3 Cõu 43 I = 3t dt dx t t = x I = t t = x + I = dx t t = + x I = t2 t (t 1) 3 dt = dt t (t 1) Bieõn soaùn: Thay Tran Sú Tuứng - Trang Bi Nguyờn hm - Tớch phõn cú li gii = dt = Cõu 48 I = 2 t3 du = 3dt t4 = t t4 dt t t t t u = 2 t t I= u www.mathvn.com du = dt 2 u du 1 u3 = 1 = u3 = x4 dx x x x +1 t t = x + I = (t 1)2 t2 dt = t 2t + t2 2 3 2t dt = t dt + 2 dt = 19 4+ + ln Dng 2: i bin s dng x ln + x ( ) dx 1+ x Cõu 49 I = x 1 x Tớnh H = 1+ x dx t x = cos t; t 0; H = u = ln(1 + x ) Tớnh K = x ln(1 + x )dx t dv = xdx Cõu 50 I = (x K= + x ) x dx I= (x + x ) x dx = x x dx + x x dx = A + B x x dx t t = x Tớnh c: A = x x dx t x = 2sin t Tớnh c: B = 2 + Tớnh B = 2 + Tớnh A = 2 Vy: I = Bieõn soaùn: Thay Tran Sú Tuứng - Trang www.MATHVN.com (3 Cõu 51 I = Bi Nguyờn hm - Tớch phõn cú li gii ) x dx 2x4 Ta cú: I = 2x + Tớnh I1 = 2x + Tớnh I = I2 = 2x4 x2 2x4 dx x dx = 21 16 dx = dx t x = 2sin t dx = cos tdt 2 x2 dx 6 cos tdt 12 = cot t dt = cot t.d (cot t ) = sin t 8 sin t Vy: I = 1( 3) 16 x 2dx x6 Cõu 52 I = t t = x dt = x dx I = 1 dt t 16 t t = 2sin u, u 0; dt = cos udu I = dt = 30 18 Cõu 53 I = x dx x+2 x 2dx Cõu 54 I = Ta cú: I = I = Cõu 55 t + 2x x2 t x = cos t dx = 2sin tdt I = sin2 dt = x 2dx 22 ( x 1)2 t x = cos t (1 + cos t ) 2sin t (2 cos t )2 dt = ( + cos t + cos2t ) dt = + 3 2 x x dx t x = sin t I = (cos t sin t )cos tdt = Bieõn soaùn: Thay Tran Sú Tuứng - Trang 12 + 8 Bi Nguyờn hm - Tớch phõn cú li gii www.mathvn.com Dng 3: Tớch phõn tng phn Cõu 56 I = x 1dx x dx u = x du = t x dv = dx v = x I = x x2 =5 I= x x 1dx x x2 dx = dx x2 x + dx x = I ln x + x ln ( + 1) + ln 2 Chỳ ý: Khụng c dựng phộp i bin x = vỡ 2;3 [ 1;1] cos t TP3: TCH PHN HM S LNG GIC Dng 1: Bin i lng giỏc Cõu 57 I = 8cos2 x sin x dx sin x cos x (sin x cos x )2 + cos x I = dx = ( sin x cos x 4(sin x + cos x ) dx sin x cos x = 3cos x 5sin x + C cot x tan x tan x dx Cõu 58 I = sin x cot x tan x cot x cos x Ta cú: I = dx = dx = dx = +C sin x sin x 2sin x sin x cos2 x + Cõu 59 I = dx sin x + cos x + + cos x + dx Ta cú: I = 2 + sin x + cos x + dx dx + = 2 + sin x + sin x + + cos x + 8 Bieõn soaùn: Thay Tran Sú Tuứng - Trang 10 www.MATHVN.com Bi Nguyờn hm - Tớch phõn cú li gii e3 e I = dx ln xdx = 3ln 4e3 + 2e2 x (1 ln x ) e2 e2 e2 Cõu 145 I = ln x ln x + x2 dx 2 t 2t + t 1 t t dx t : t = ln x dt = I = dt = dt = dt + dt = I1 + I t t t x e e e et tdt dt dt dt + I1 = = tet + = t t 0 t e e et e e + I2 = tdt dt t et e 2(e 1) Vy : I = 2 dt 1 = tet + et 2 = tet = t e e2 e dt e2 ln( x + 1) Cõu 146 I = dx x x 1+ t t = ln ( x + 1) 2dt = e3 Cõu 147 I = dx x 1+ x ln dt = ln2 ln2 ln ln x dx x + ln x t t = + ln x + ln x = t (t 1)3 I = dt = t Cõu 148 I = I =2 e ln x x + ln x t 3t + 3t 1 15 dt = (t 3t + 3t )dt = ln t t 1 dx t t = + ln x I = e (2 t )dt = xe x + 1 x(e x + ln x) dx 3 t t = + ln2 x I = 34 24 e Cõu 150 I = ln x + ln2 x Cõu 149 I = dx x e dx = 2tdt v ln3 x = (t 1)3 x t t = e x + ln x I = ln ee + e Bieõn soaùn: Thay Tran Sú Tuứng - Trang 29 Bi Nguyờn hm - Tớch phõn cú li gii www.mathvn.com Dng 2: Tớch phõn tng phn Cõu 151 I = e s inx sin xdx u = sin x du = cos xdx sin x cos xdx v = esin x dv = e I = es inx sin x cos xdx t I = 2sin xesin x 02 e sin x cos xdx = 2e 2esin x 02 =2 Cõu 152 I = x ln( x + x + 1)dx 2x + du = dx u = ln( x + x + 1) x + x +1 t dv = xdx v = x x2 1 2x3 + x2 I= ln( x + x + 1) dx 2 x + x +1 3 11 1 2x + 31 dx = ln (2 x 1)dx + dx = ln 2 20 x + x +1 x + x +1 12 Cõu 153 I = ln x x +1 dx u = ln x dx 8 x +1 du = ( ) dx I = x + 1.ln x x x dx = ln ln 2J x + v = x + t dv = + Tớnh J = 3 3 x +1 t t2 1 dx t t = x + J = dt = + 2tdt = dt 2 x t t + 1 t t 1 2 Bieõn soaùn: Thay Tran Sú Tuứng - Trang 30 www.MATHVN.com Bi Nguyờn hm - Tớch phõn cú li gii t = 2t + ln t +1 = + ln ln T ú I = 20 ln ln x + x ln x + x e dx x e Cõu 154 I = e e e x e dx x I = xe x dx + ln xe x dx + 1 e e e x e e 1 e x e e dx = ee dx x x 1 +Tớnh I = e x ln xdx = e x ln x e + Tớnh I1 = xe x dx = xe x e x dx = ee (e 1) e x e dx = ee +1 x Vy: I = I1 + I + e ln x + ln2 x dx x + ln x Cõu 155 I = e ln x 1x e + ln x Tớnh I1 = dx t t = + ln x I1 = 2 3 + Tớnh I = ln2 xdx Ly tớch phõn tng phn ln c I = e 2 3 ln( x + 1) Cõu 156 I = dx x Vy I = e 2x u = ln( x + 1) du = ln( x + 1) 2 dx x + t Do ú I = + dx 2 dv = x v = x ( x + 1) x 2x2 = ln ln x ln ln dx d ( x + 1) + dx = + x x2 + x x + = ln ln + ln | x | ln | x + 1| = ln ln Cõu 157 I = ln( x + 1) x2 dx dx u = ln( x + 1) du = x + 1 dx dx t I = ln( x + 1) + = 3ln ln dv = 1 ( x + 1) x x v = x2 x 1+ x dx x Cõu 158 I = x ln Bieõn soaùn: Thay Tran Sú Tuứng - Trang 31 Bi Nguyờn hm - Tớch phõn cú li gii www.mathvn.com du = dx 1+ x 1+ x 2 (1 x )2 t u = ln x I = x ln x dx 2 x x x dv = xdx v = 0 = 2 ln x ln ln 1 + dx = + + dx = + + ln x ( x 1)( x + 1) 2 10 u = ln x + t x I = ln ln + dv = x 2dx Cõu 159 I = x ln x + dx x 2 t u = ln(12 + x ) I = ln + + Cõu 160 I = x ln(1 + x )dx ln x Cõu 161 I = ( x + 1) u = ln x dx t dv = ( x + 1) dx I = ln + ln ln x + e x (e x + ln x) dx + ex e Cõu 162 I = dv = x dx e e Ta cú: I = ln x.dx + e2 x 1e x +1 dx = H + K e e + H = ln2 x.dx t: u = ln x H = e ln x.dx = e dv = dx 1 e2 x e + K= 1e x dx t t = e + I = x +1 Vy: I = ee + ln x e +1 Ta cú: I = e x+ x t e +1 dt = ee e + ln t ee + e +1 ee + Cõu 163 I = ( x + )e ee +1 x+ x dx 1 x+ dx + x e x dx = H + K x + Tớnh H theo phng phỏp tng phn I1 = H = xe x+ x I= 1 x+ x 52 x e dx = e K x 2 e Cõu 164 I = ln( x + x)dx Bieõn soaùn: Thay Tran Sú Tuứng - Trang 32 www.MATHVN.com Bi Nguyờn hm - Tớch phõn cú li gii ( ) t u = ln x + x I = x ln dv = dx ( x + x) 4 + x x2 + dx = TP5: TCH PHN T HP NHIU HM S Cõu 165 I = x 2e x + x dx 1+ x I = x 2e x dx + x 1+ x dx 11 t 1 1 + Tớnh I1 = x e dx t t = x I1 = e dt = et = e 30 3 x3 + Tớnh I = 1+ dx t t = x I = x Vy: I = e + 3 x2 Cõu 166 I = x e x x3 2 x I = xe dx + x2 dx 2 + Tớnh I1 = xe x dx = e2 + Tớnh I = I2 = cos2 t dt = ( cot t t ) = sin t Cõu 167 I = 4x + Tớnh I1 = x e + Tớnh I = I= 2x ) 4x x2 dx t x = 2sin t , t 0; x x dx x3 I = x e2 x dx x2 ( e2 x x Vy: I = e2 + dx x2 t4 dt = + 1+ t x dx = I1 + I e2 + dx = x3 x2 dx t t = x I = 3 + 16 e2 61 +3 12 Bieõn soaùn: Thay Tran Sú Tuứng - Trang 33 Bi Nguyờn hm - Tớch phõn cú li gii x2 + Cõu 168 I = ( x + 1) e x dx 2 t t = x + dx = dt I = t 2t + t2 Cõu 169 I = x +1 x e www.mathvn.com e2 2 et 1dt = + et 1dt = e + + e = e t2 t dx + x2 2 t t = + x dx = tdt I = (t 1)e dt = t 2et dt et t 1 = J ( e e) 2 + J = t e dt = t e 2te dt = 4e e tet et dt = 4e2 e 2(tet et ) 1 1 1 2 t t 2 t Vy: I = e2 x ln( x + 1) + x Cõu 170 I = x2 + Ta cú: f ( x ) = dx x ln( x + 1) + x ( x + 1) x = x ln( x + 1) +x x x +1 x +1 x +1 x +1 1 F ( x ) = f ( x )dx = ln( x + 1)d ( x + 1) + xdx d ln( x + 1) 2 2 = ln ( x + 1) + x ln( x + 1) + C 2 ( 4 I = ) ln x + x + x Cõu 171 I = x2 + ( ) ln x + x + x x +9 + Tớnh I1 = I1 = ln ( x +9 udu = ln + Tớnh I = Vy I = ( ln x + x + x +9 )dx t ln ( x + )dx 34 x3 x +9 ) x + = u du = dx = I1 3I 2 x +9 u2 ln ln2 ln2 = ln x3 x +9 x + = v dv = dx t I = (u2 9)du = ( dx = ln x + x + dx x x +9 dx , x = v u 44 9u) = 3 ( ) ln x + x + x x2 + ln ln dx = I1 3I = 44 ( x + 1) ln x + x + dx + x ln x e Cõu 172 I = Bieõn soaùn: Thay Tran Sú Tuứng - Trang 34 dx www.MATHVN.com e Bi Nguyờn hm - Tớch phõn cú li gii e x3 e3 + x dx = = 3 e e + ln x I = x 2dx + dx + x ln x 1 e + e + ln x d (2 + x ln x ) dx = + x ln x + x ln x = ln + x ln x 1 Cõu 173 I = e3 x ln3 x + ln x e = ln e+2 Vy: I = e3 e+2 + ln dx t t = + ln x + ln x = t (t 1)3 dt = t I = dx = 2tdt v ln3 x = (t 1)3 x t 3t + 3t 1 15 dt = (t 3t + 3t )dt = ln t t 1 Cõu 174 I = x sin x dx x cos u= x t sin x = dv dx cos2 x dx cos xdx t t = sin x I1 = cos x = sin2 x 0 + I1 = du = dx x I= cos x v = cos x 2 dx dx = cos x cos x dt t = 2+ ln 2 2+ ln 2 Vy: = ln(5 x) + x3 x Cõu 175 I = dx x2 ln(5 x) Ta cú: I = dx + x x dx = K + H x2 1 4 u = ln(5 x ) ln(5 x ) dx dx t K = ln = dv x x2 + K= + H= x x dx t t = x H = 164 15 164 Vy: I = ln + 15 Cõu 176 I = x (2 x ) + ln(4 + x ) dx 2 Ta cú: I = x(2 x )dx + ln(4 + x )dx = I1 + I 2 + I1 = x (2 x )dx = ( x 1)2 dx = 0 (s dng i bin: x = + sin t ) Bieõn soaùn: Thay Tran Sú Tuứng - Trang 35 Bi Nguyờn hm - Tớch phõn cú li gii 2 2 + I = ln(4 + x )dx = x ln(4 + x ) www.mathvn.com x2 dx (s dng tớch phõn tng phn) + x2 = ln + (i bin x = tan t ) 0 Vy: I = I1 + I = + ln 2 ln x dx x +1 Cõu 177 I = u = ln x dx 8 x +1 du = dx t I = x + ln x dx x dv = x x + v = x + 3 x +1 2t dt + Tớnh J = dx t t = x + J = = + dt = + ln ln 2 x t t I = ln ln 2(2 + ln ln 2) = 20 ln ln + x2 x3 ln xdx Cõu 178 I = u = ln x 1 1 Ta cú: I = + ln xdx t dv = ( + )dx x x x3 x 2 1 63 I = + ln x ln x + ln x dx = ln + + ln2 x 64 4x 4x x + x ln x + x e dx x e Cõu 179 I = e e 1 e x e dx = H + K + J x Ta cú: I = xe x dx + e x ln xdx + e e e + H = xe x dx = xe x 1e e x dx = ee (e 1) e x e e e + K = e ln xdx = e ln x dx = e dx = ee J x x 1 x x e e x Vy: I = H + K + J = ee +1 ee + ee J + J = ee +1 Cõu 180 I = x cos x sin3 x dx cos x Ta cú = t sin x sin x 1 I = x 2 sin x u = x du = dx dv = cos x dx v = sin3 x 2sin x 2 dx 1 + = ( ) cot x = 2 sin x 2 2 4 Bieõn soaùn: Thay Tran Sú Tuứng - Trang 36 www.MATHVN.com Bi Nguyờn hm - Tớch phõn cú li gii Cõu 181 I = cos3 x dx x sin x u = x du = dx x sin x t: I = dv = dx v= cos2 x cos3 x 2.cos2 x dx cos2 x = tan x 0 = ( x + sin x) + sin x dx Cõu 182 I = x sin2 x dx + + sin x + sin x dx = H + K 0 Ta cú: I = u = x du = dx dx x x = dv + H= dx = dx t: v = tan x + sin x 0 cos2 x cos x 2 H= x 2 tan x + ln cos x = + K= 2 sin x cos2 x dx t t = x K = + sin x + sin x dx 0 dx = tan x = K = cos2 x Vy, I = H + K = + 2K = Cõu 183 I = x (cos3 x + cos x + sin x ) + cos2 x dx cos x (1 + cos2 x ) + sin x x.sin x Ta cú: I = x dx = J + K dx = x.cos x.dx + 2 + cos x 0 + cos x u = x + Tớnh J = x.cos x.dx t J = ( x.sin x ) sin x.dx = + cos x = 0 dv = cos xdx 0 + Tớnh K = x.sin x + cos x K= dx t x = t dx = dt ( t ).sin( t ) 2K = + cos ( t ) ( x + x ).sin x + cos2 x dt = ( t ).sin t + cos t dx = dt = sin x.dx + cos x ( x ).sin x + cos2 x K= dx sin x.dx + cos2 x Bieõn soaùn: Thay Tran Sú Tuứng - Trang 37 Bi Nguyờn hm - Tớch phõn cú li gii t t = cos x dt = sin x.dx K = K= Vy I = (1 + tan u)du Cõu 184 I = x + ( x + sin x )sin x (1 + sin x )sin x Ta cú: I = 3 + K = Vy I = x sin x u = 4 dx dx = x sin x dx + dx =H+K + sin x u = x du = dx H= dv = dx v = cot x sin x dx t t t = tan u dt = (1 + tan2 u)du (1 + sin x )sin x dx = + sin x Cõu 185 I = du = x (1 + sin x ) + sin x + H = dt + t2 , = + tan u www.mathvn.com dx dx = = x + cos x cos + 32 x + sin2 x dx + cos2 x x + sin x x sin x Ta cú: I = 03 dx = dx + dx = H + K 0 + cos2 x cos2 x cos2 x u = x x x du = dx dx + H= dx = dx t 2 dv = cos x v = tan x cos x cos2 x H = x tan x tan xdx = + ln cos x = ln 0 2 sin x 1 + K = dx = tan2 xdx = [ tan x x ] = 0 2 cos2 x Vy: I = H + K = 1 ( 1) ln + = + ( ln 2) 2 Cõu 186 I = x + 1sin x + 1.dx 2 1 t t = x + I = t.sin t.2tdt = 2t sin tdt = x sin xdx Bieõn soaùn: Thay Tran Sú Tuứng - Trang 38 www.MATHVN.com Bi Nguyờn hm - Tớch phõn cú li gii 2 u = x du = xdx t I = x cos x + x cos xdx dv = sin xdx v = cos x u = x du = 4dx t T ú suy kt qu dv = cos xdx v = sin x + sin x + cos x e Cõu 187 I = x dx I= 2 e dx sin x x + e dx + cos x x cos x x x 2sin cos sin x x 2 e x dx = tan x e x dx + Tớnh I1 = e dx = x + cos x 0 cos2 u = ex du = e x dx x 2 e dx tan x e x dx + Tớnh I = t dv = dx I = e 2 x x 20 v = tan x cos2 cos 2 2 Do ú: I = I1 + I = e Cõu 188 I = I = 02 I = cos x e x (1 + sin x ) dx cos x (sin x + cos x )dx u= du = x cos x e ex dx t x dx sin x e (sin x + cos x ) dv = v = (sin x + cos x ) sin x + cos x cos x e x sin x sin x + cos x + sin xdx e x = sin xdx ex u1 = sin x du1 = cos xdx t I = sin x x + dx e dv1 = x v1 = x e e u2 = cos x du2 = sin xdx t dx dv1 = x v1 = x e e I = e2 + cos x ex sin xdx ex = + I 2I = cos xdx e e x = e2 +1 I = e2 Bieõn soaùn: Thay Tran Sú Tuứng - Trang 39 + cos xdx ex e 2 + Bi Nguyờn hm - Tớch phõn cú li gii Cõu 189 I = sin x + cos6 x 6x + dx t t = x dt = dx I = I = (6 x + 1) 6x + 6 sin t + cos t 6t + dt = 6x dx = sin x + cos6 x 6x + (sin x + cos6 x )dx = + cos x dx = 16 8 32 Cõu 190 I = sin xdx x + Ta cú: I = sin xdx 2x + + Tớnh I1 = x x sin xdx 2x + x sin xdx 2x + + x sin xdx 2x + sin xdx 2x + = I1 + I 2t sin (t ) t + t x = t I1 = 6 = dt = + sin xdx 2x + x = sin xdx = 0 sin t 6 16 (1 cos2 x )2 dx 40 e cos(ln x )dx t t = ln x x = et dx = et dt I = et cos tdt = (e + 1) (dựng pp tớch phõn tng phn) 2 Cõu 192 I = esin x sin x.cos3 xdx t t = sin x I = 11 t e (1 t )dt = e (dựng tớch phõn tng phn) 20 Bieõn soaùn: Thay Tran Sú Tuứng - Trang 40 sin x 2t + 1dt = x + 1dx 16 (3 cos x + cos x )dx = 80 64 Cõu 191 I = = dx sin x + cos x I = 6t 2I = www.mathvn.com www.MATHVN.com Bi Nguyờn hm - Tớch phõn cú li gii Cõu 193 I = ln(1 + tan x )dx t t = x I = ln + tan t dt = = ln 2dt ln(1 + tan t )dt 2I = tan t ln + + tan t dt = ln + tan t dt 4 ln I = = t.ln 04 I ln Cõu 194 I = sin x ln(1 + sin x )dx u = ln(1 + sin x ) t dv = sin xdx + cos x du = + sin x dx v = cos x I = cos x.ln(1 + sin x ) + cos x 0 2 cos x sin x dx = + dx = (1 sin x )dx = 1 + sin x + sin x 0 Cõu 195 I = tan x.ln(cos x ) dx cos x t t = cos x dt = sin xdx I = ln t t dt = ln t t2 dt u = ln t du = t dt t I = ln dv = dt 2 v= t t TP6: TCH PHN HM S C BIT Cõu 196 Cho hm s f(x) liờn tc trờn R v f ( x ) + f ( x ) = cos4 x vi mi x R Tớnh: I= f ( x )dx t x = t f ( x )dx = f (t )( dt ) = f (t )dt = f ( x )dx Bieõn soaùn: Thay Tran Sú Tuứng - Trang 41 Bi Nguyờn hm - Tớch phõn cú li gii 2 www.mathvn.com f ( x )dx = f ( x ) + f ( x ) dx = 2 cos4 xdx I = 16 1 + cos2 x + cos x 8 Chỳ ý: cos4 x = Cõu 197 Cho hm s f(x) liờn tc trờn R v f ( x ) + f ( x ) = + cos2 x , vi mi x R I= Tớnh: f ( x )dx 3 Ta cú : I = f ( x )dx = 3 f ( x )dx (1) f (t )dt = f ( x )dx Thay vo (1) ta c: I = = cos xdx f ( x )dx t x = t dx = dt I1 = f ( x )dx + + Tớnh : I1 = 3 f ( x ) + f ( x ) dx = cos xdx = sin x 02 sin x (1 + cos x ) = cos x dx =6 Cõu 198 I = sin x + x2 + x dx I= + x sin xdx x sin xdx = I1 I + Tớnh I1 = + x sin xdx S dng cỏch tớnh tớch phõn ca hm s l, ta tớnh c I1 = + Tớnh I = x sin xdx Dựng pp tớch phõn tng phn, ta tớnh c: I = Suy ra: I = Bieõn soaùn: Thay Tran Sú Tuứng - Trang 42 + www.MATHVN.com Cõu 199 I = I = Bi Nguyờn hm - Tớch phõn cú li gii e x (3x 2) + x e x ( x 1) + x e x ( 3x ) + x dx e x ( x 1) + x + e x ( x 1) e x ( x 1) 5 x x e ( x 1) + x u= x x cos x cos x dx t x cos x cos x ( x sin x + cos x ) dx dv = ( x sin x + cos x )2 cos x + x sin x dx du = cos x v = x sin x + cos x e x ( x 1) + x dx = e x ( x 1) + x dx = dx + e ( x 1) e ( x 1) =x + dx = + dx x 2 x 1(e x + 1) x 1(e x x + 1) e x ( x 1) t t = e x x + dt = dx x x I = 3+ e5 +1 e2 +1 Cõu 200 I = x 2e + 2e + dt I = + ln t = + ln t e +1 e +1 x2 ( x sin x + cos x )2 dx I= I = x + cos x ( x sin x + cos x ) dx cos2 x dx = 4+ Bieõn soaùn: Thay Tran Sú Tuứng - Trang 43 dx [...]... 1005ln(ln( x 2 + 1) + 1) + C 2 t 2 2 2 Cõu 124 I = e Cõu 125 J = 1 xe x + 1 x (e x + ln x ) dx e d (e x + ln x ) 1 e x + ln x J= = ln e x + ln x Bieõn soaùn: Thay Tran Sú Tuứng - Trang 25 e 1 = ln ee + 1 e Bi tp Nguyờn hm - Tớch phõn cú li gii ln 2 Cõu 126 I = e3 x + e 2 x e x + 1 0 I= ln 2 2 e3 x + e 2 x 1 = ln(e3 x + e2 x 2 x 3 e dx 3ln 2 ( x 0 3 x 3 e e Cõu 128 I = ln 2 dx = dx ( 3 ex + 2) ln... Cõu 120 I = 4 Bi tp Nguyờn hm - Tớch phõn cú li gii x cos 2 x (1 + sin 2 x ) 0 2 dx u = x du = dx cos2 x t 1 dv = dx = v 2 1 + sin 2 x (1 + sin 2 x ) 4 4 1 1 1 1 1 dx = + 4+ 16 2 2 1 + sin 2 x 0 2 0 1 + sin 2 x I = x 1 1 2 cos2 x 4 0 dx 1 1 1 2 2 = + tan x 4 = + 0 + 1) = ( 16 2 2 4 16 2 2 4 16 0 TP4: TCH PHN HM S M - LOGARIT Dng 1: i bin s Cõu 121 I... Cõu 121 I = e2 x 1 + ex dx t t = e x e x = t 2 e x dx = 2tdt t3 2 2 I = 2 dt = t 3 t 2 + 2t 2 ln t + 1 + C = e x e x e x + 2 e x 2 ln e x + 1 + C 1+ t 3 3 Cõu 122 I = I = ( x 2 + x )e x x + e x ( x 2 + x )e x x x+e dx Cõu 123 I = dx dx = xe x ( x + 1)e x xe + 1 x dx t t = x.e x + 1 I = xe x + 1 ln xe x + 1 + C e2 x + 9 t t = e2 x + 9 I = dt t2 9 = 1 t 3 1 ln + C = ln 6 t+3 6 e2... x I = 64 16 64 128 Cõu 63 I = 2 cos2 x(sin 4 x + cos4 x )dx 0 2 0 1 1 2 0 1 I = cos2 x 1 sin2 2 x dx = 1 sin2 2 x d (sin 2 x ) = 0 2 2 2 Cõu 64 I = 2 3 (cos x 1) cos2 x.dx 0 Bieõn soaùn: Thay Tran Sú Tuứng - Trang 11 = Bi tp Nguyờn hm - Tớch phõn cú li gii 5 cos xdx = 2 (1 sin x ) 2 A = 0 2 d (sin x ) = 0 8 15 2 2 cos x.dx = B= 2 www.mathvn.com 0 12 (1 + cos 2 x ).dx... 0 I= dx 3e3 x + 2e2 x e x (e3 x + e2 x e x + 1) 0 Cõu 127 I = www.mathvn.com +2 ) 2 t t x = e3 x 1 3 3 1 dt = e 3 dx I = ln 3 4 2 6 3 x e 1 dx 0 t 3 x e 1 = t dx = 1 3t 2 dt t3 + 1 1 I = 3 1 0 1 1 dt dt = 3 3 3 t3 + 1 t + 1 0 1 1 2t + = + ln 2 dt = 3 2 t + 1 3 t + 1 t t + 1 0 0 Tớnh I1 = 3 dt Vy: I = 3 ln 2 Cõu 129 I = 3 ln15 ( e2 x 24e x ) dx 3ln 2 e x e x + 1 + 5e... 0 3 sin x 3 cos x dx + 2 sin x 3 cos x dx = 3 3 0 3 Cõu 86 I = 2 sin xdx (sin x + cos x )3 0 t x = 2 t dx = dt I = 2 cos tdt = cos xdx (sin t + cos t )3 (sin x + cos x )3 0 2 0 2 12 dx 1 4 1 2I = = = cot( x + ) = 1 I = 2 20 2 2 4 0 2 0 (sin x + cos x ) sin ( x + ) 4 dx Cõu 87 I = 2 7sin x 5cos x (sin x + cos x )3 dx 0 Xột: I1 = t x = 2 2 0 sin xdx ( sin x + cos x... 4 sin t cos t cos3 t + sin3 t dt = 2 sin 4 x cos x cos3 x + sin3 x dx 0 2 2 2I = 4 4 cos x sin x + sin x cos x sin3 x + cos3 x 0 dx = 2 0 3 3 sin x cos x (sin x + cos x ) sin3 x + cos3 x dx = 12 1 sin 2 xdx = 20 2 1 4 I= 2 0 1 cos2 (sin x ) tan Cõu 91 I = t x = 2 2 (cos x ) dx t dx = dt 2 2 1 tan 2 (sin t ) dt = tan 2 (sin x ) dx 2 2 cos (cos t ) cos (cos x ) 0 0... Bieõn soaùn: Thay Tran Sú Tuứng - Trang 17 1 2 Bi tp Nguyờn hm - Tớch phõn cú li gii t u = sin x + cos x I = 2 1 www.mathvn.com du 4u 2 t u = 2sin t I = 4 2 cos tdt 2 4 4sin t 6 4 = dt = 12 6 Cõu 93 I = 3 sin x 2 cos x 3 + sin x 0 4 cos2 x Ta cú: cos2 x = 4 t 2 v dt = t t = 3 + sin2 x = I= 3 3 0 = dx sin x dx = cos x 3 + sin2 x 15 2 1 t+2 ln 4 t2 3 2 3 2 sin x 2 3 + Tớnh I1 =... + cos x I = 2 4 21 I = 1 t 2 2 dt = ln t 1 = 1 ln 2 2 6 Cõu 103 I = 2 1 cos3 x sin x.cos5 xdx 1 t t = 6 1 cos3 x t 6 = 1 cos3 x 6t 5dt = 3cos2 x sin xdx dx = 2t 5dt cos2 x sin x 1 1 t 7 t13 12 I = 2 t 6 (1 t 6 )dt = 2 = 7 13 0 91 0 Cõu 104 I = 4 tan xdx 0 cos x 1 + cos2 x Ta cú: I = 3 4 tan xdx 0 cos2 x tan2 x + 2 tdt I= = t 2 t t = 2 + tan 2 x t 2 = 2 + tan 2 x tdt = 3... 4 4 3 sin x 3 cos x t t = tan x I = dx = cos8 x 3 3 t 4 dt 3 4 4 1 1 2 tan x cos x 3 dx = 4 ( 8 3 1) 1 Bieõn soaùn: Thay Tran Sú Tuứng - Trang 21 6 2 Bi tp Nguyờn hm - Tớch phõn cú li gii Cõu 112 I = x( 0 www.mathvn.com cos3 x + cos x + sin x )dx 1 + cos 2 x cos x (1 + cos2 x ) + sin x x.sin x dx = x.cos x.dx + dx = J + K 2 2 1 + cos x 1 + cos x 0 0 Ta cú: I = x 0 u = x du ... x + 1) dt x dx 128 t I = dx t t = x I = dt 7 t (1 + t ) x (1 + x ) dx (1 x ).x dx x (1 + x ) t : x = I = t 3 t6 dt = t2 + 1 117 41 + t t +1 dt = 135 12 t + x 2001 Cõu 14 I =... (1 + x )1002 x 2004 I = dx 1002 x (1 + x ) Cỏch 2: Ta cú: I = dx = 1 1002 x + x 1000 + x2 11+ x Ta cú: 1+ x 1+ x4 x2 + dt = x3 dx 11 x 2000 xdx t t = + x dt = xdx 2000 2 (1 + x )... 3dt = 2 2011 2011x dx = x2 3 = 213 128 14077 16 14077 21 16 128 dx Cõu 47 I = (1 + x ) + x 3 3 = 15 ln 2 2t dt = + dt = + ln 12 t t2 + t2 + t(t + 1) Cõu 44 I = t3