数学分析课后答案(复旦大学-第三版)ch-6-12.pdf

104 1C 18 1.Vg9$K 1. yeRf(t)dt = F(t) + CKRf(ax + b)dx = 1 aF(ax + b) + C. yRf(t)dt = F(t) + CF(t) + C0= f(t)K ? 1 aT(ax + b) ?0 = 1 aF(ax + b) 0 = f(ax + b)u Rf(ax + b)dx = 1 aF(ax + b) + C. 2. e (1) Z (2 sec2x)dx (2) Z ? x4 2x3+ x 2 ? dx (3) Z ? x + 3 x + 2 x+ 2 3 x 2 ? dx (4) Z ? ex+ 1 x + 1 x2 + 1 x3 ? dx (5) Z ? 2cosx + 1 2 sinx ? dx (6) Z ? cosx 2 1 + x2 + 1 41 x2 ? dx (7) Z ?1 2 cosx + sinx + 1 ? dx (8) Z ? 2x+ ? 1 3 ?x ex 5 ? dx (9) Z (3 x2)3dx (10) Z ? 1 1 x2 ?q xxdx ) (1) Z (2 sec2x)dx = 2x tanx + C (2) Z ? x4 2x3+ x 2 ? dx = 1 5x 5 1 2x 4 + 1 3x 3 2+ C (3) Z ? x + 3 x + 2 x+ 2 3 x 2 ? dx = 2 3x 3 2+ 3 4x 4 3 2x + 3x 2 3+ 4x 1 2+ C (4) Z ? ex+ 1 x + 1 x2 + 1 x3 ? dx = ex+ ln|x| 1 x 1 2x2 + C (5) Z ? 2cosx + 1 2 sinx ? dx = 2sinx 1 2 cosx + C (6) Z ? cosx 2 1 + x2 + 1 41 x2 ? dx = sinx 2arctanx + 1 4 arcsinx + C (7) Z ?1 2 cosx + sinx + 1 ? dx = 1 2 sinx cosx + x + C (8) Z ? 2x+ ? 1 3 ?x ex 5 ? dx = 1 ln22 x 1 ln3 ? 1 3 ?x ex 5 + C (9) Z (3 x2)3dx = Z (27 27x2+ 9x4 x6)dx = 27x 9x3+ 9 5x 5 1 7x 7 + C (10) Z ? 1 1 x2 ?q xxdx = Z (x 3 4 x 5 4)dx = 4 7x 7 4+ 4x 1 4+ C 105 2.O 1. e (1) Z dx 5x 7 (2) Z cos(t )dt (3) Z dx r 1 ?x 2 + 3 ?2 (4) Z dx 1 2x2 (5) Z tan10xsec2xdx (6) Z ex 2xdx (7) Z (2x+ 3x)2dx (8) Z tanxdx (9) Z tan p 1 + x2 xdx 1 + x2 (10) Z (x2+ )xdx( 6= 1) (11) Z dx 1 cosx (12) Z dx A2sin2x + B2cos2x (13) Z sinx cosx 1 + sin4x dx (14) Z dx sin2 ? x + 4 ? (15) Z x2 8 p 1 + x3dx (16) Z sin2xcosx 1 + sin3x dx (17) Z 1 2sinx cos2x dx (18) Z dx ex+ ex (19) Z sinx + cosx 3 sinx cosxdx (20) Z 1 + sin2x sin2x dx (21) Z s ln(x + 1 + x2) 1 + x2 dx (22) Z dx 1 + e2x (23) Z dx x2 2x + 2 (24) Z dx (arcsinx)21 x2 106 (25) Z x2+ 7 x2 2x 3 dx (26) Z x2 1 x4+ 1 dx ) (1) Z dx 5x 7 = 1 5 Z d(5x 7) 5x 7 = 1 5 ln|5x 7| + C (2) Z cos(t )dt = 1 Z cos(t )d(t ) = 1 sin(t ) + C (3) Z dx r 1 ?x 2 + 3 ?2 = 2 Z d ?x 2 + 3 ? r 1 ?x 2 + 3 ?2 = 2arcsin ?x 2 + 3 ? + C (4) Z dx 1 2x2= 2 2 Z d(x) q 1 (2x)2 = 2 2 arcsin(2x) + C (5) Z tan10xsec2xdx = Z tan10xd(tanx) = 1 11 tan11x + C (6) Z ex 2xdx = Z (2e)xdx = (2e)x ln(2e) + C (7) Z (2x+ 3x)2dx = Z (4x+ 2 6x+ 9x)dx = 4x ln4 + 2 ln66 x + 9x ln9 + C (8) Z tanxdx = Z sinx cosx dx = Z d(cosx) cosx = ln|cosx| + C = ln|secx| + C (9) Z tan p 1 + x2 xdx 1 + x2= Z tan p 1 + x2d( p 1 + x2) = ln|sec p 1 + x2| + C (10) Z (x2+ )xdx = 1 2 Z (x2+ )d(x2+ ) = (x2+ )+1 2( + 1) + C (11) Z dx 1 cosx = Z csc2 x 2 d ?x 2 ? = cot x 2 + C (12) Z dx A2sin2x + B2cos2x = 1 AB Z 1 1 + ? A B ?2 tan2x dAtanx B = 1 AB arctan ? A B tanx ? + C (13) Z sinx cosx 1 + sin4x dx = 1 2 Z 1 1 + (sin2x)2 d(sin2x) = 1 2 arctan(sin2x) + C (14) Z dx sin2 ? x + 4 ? = Z csc2 ? x + 4 ? d ? x + 4 ? = cot ? x + 4 ? + C (15) Z x2 8 p 1 + x3dx = 1 3 Z 8 p 1 + x3d(1 + x3) = 8 27(1 + x 3)9 8+ C (16) Z sin2xcosx 1 + sin3x dx = 1 3 Z d(1 + sin3x) 1 + sin3x = 1 3 ln(1 + sin3x) + C (17) Z 1 2sinx cos2x dx = Z sec2xdx + 2 Z dcosx cos2x = tanx 2secx + C (18) Z dx ex+ ex = Z dex e2x+ 1 = arctan(ex) + C (19) Z sinx + cosx 3 sinx cosxdx = Z d(sinx cosx) 3 sinx cosx= 3 2(sinx cosx) 2 3+ C (20) Z 1 + sin2x sin2x dx = Z csc2xdx + Z d(sin2x) sin2x = cotx + ln(sin2x) + C = cotx + 2ln|sinx| + C (21) Z s ln(x + 1 + x2) 1 + x2 dx = Z q ln(x + p 1 + x2)d(ln(x + p 1 + x2) = 2 3ln(x + p 1 + x2 3 2+ C 107 (22) Z dx 1 + e2x= Z dex 1 + e2x= ln(ex+ p 1 + e2x) + C (23) Z dx x2 2x + 2 = Z d(x 1) (x 1)2+ 1 = arctan(x 1) + C (24) Z dx (arcsinx)21 x2 = Z d(arcsinx) (arcsinx)2 = 1 arcsinx + C (25) Z x2+ 7 x2 2x 3 dx = Z ? 1 + 2x + 10 (x + 1)(x 3) ? dx = Z ? 1 2 x + 1 + 4 x 3 ? dx = x 2ln|x + 1| + 4ln|x 3| + C = x + 2ln (x 3)2 |x + 1| + C (26) Z x2 1 x4+ 1 dx = Z 1 x2 x2+ x2 dx = Zd ? x + 1 x ? ? x + 1 x ?2 2 = 2 4 ln ? ? ? ? ? ? ? x + 1 x 2 x + 1 x + 2 ? ? ? ? ? ? ? + C = 2 4 ln ? ? ? ? x2 2x + 1 x2+ 2x + 1 ? ? ? ?+ C 2. e (1) Z sin x xdx (2) Z (2u + 1)2 u2 du (3) Z e x+1 dx (4) Z x2 4 x2dx (5) Z p x2+ a2dx (6) Z p x2 a2dx (7) Z dx p(x a)(b x) (8) Z dx x2px2+ (9) Z xdx 5 + x x2 (10) Z p 2 + x x2dx ) (1) Z sin x xdx = 2 Z sin xd(x) = 2cosx + C (2) Z (2u + 1)2 u2 du = Z ? 4 u + 4 u 3 2 + 1 u2 ? du = 4ln|u| 8u 1 2 1 u + C (3) -1 + x = tKx = t21, dx = 2tdtu Z e x+1 dx = 2 Z tetdt = 2(t1)et+C = 2(1 + x 1)e 1+x + C (4) Z x2 4 x2dx = Z 4 x2 4 4 x2dx = Z p 4 x2dx + 4 Z dx 4 x2= x 2 p 4 x2 2arcsin x 2 + 4arcsin x 2 + C = 2arcsin x 2 x 2 p 4 x2+ C 108 (5) I = Z p x2+ a2dx = x p x2+ a2 Z x2 x2 + a2 dx = x p x2+ a2 Z p x2+ a2dx+ Z a2 x2 + a2 dx = x p x2+ a2 I + a2ln|x + p x2+ a2| + C1 u2I = xx2+ a2+a2ln|x+ x2 + a2|+C1l I = x 2 p x2+ a2+ a2 2 ln(x+ p x2+ a2)+C(C = C1 2 ) (6) I = Z p x2 a2dx = x p x2 a2 Z x2 x2 a2 dx = x p x2 a2 Z p x2 a2dx Z a2 x2 a2 dx = x p x2 a2 I a2ln|x + p x2 a2| + C1 u2I = xx2 a2a2ln|x+ x2 a2|+C1l I = x 2 p x2 a2 a2 2 ln(x+ p x2 a2)+C(C = C1 2 ) (7) Z dx p(x a)(b x)= Z dx px2 (a + b)x ab = Zd ? x a + b 2 ? s ? x a + b 2 ?2 + ? a b 2 ?2 = arcsin x a + b 2 a b 2 + C = arcsin 2x a b a b + Ca 0) (12) Z xdx 4 px3(a x) (13) Z x p x4+ 2x2 1dx (14) Z p 2 + x x2dx (15) Z x2dx 1 + x x2 (16) Z x2+ 1 xx4+ 1 dx (17) Z sin6xdx (18) Z sin2xcos4xdx (19) Z sin4xcos4xdx (20) Z cos4x sin3x dx (21) Z dx sin3xcos5x (22) Z tanx tan(x + a)dx 112 (23) Z sin5xcosxdx (24) Z sin2x 1 + sin2x dx (25) Z dx sin(x + a)sin(x + b) (26) Z xexcosxdx (27) Z dx (2 + cosx)sinx (28) Z ln(x + p 1 + x2)2dx (29) Z sinxcosx sinx + cosx dx (30) Z lnx (1 + x2) 3 2 dx (31) Z xexsinxdx (32) Z x3arccosx 1 x2dx (33) Z (x + |x|)2dx (34) Z x2excosxdx (35) Z xex (1 + x)2 dx (36) Z xln2 xdx (37) Z dx p(x a)(b x) (38) Z xln 1 + x 1 x dx (39) Z xarctanx ln(1 + x2)dx (40) Z sinh2xcosh2xdx ) (1) -tan x 2 = tKcosx = 1 t2 1 + t2 , dx = 2dt 1 + t2 u Z dx 4 + 5cosx = Z 2 (3 t)(3 + t) dt = 1 3 ln ? ? ? ? 3 + t 3 t ? ? ? ?+ C = 1 3 ln ? ? ? ? ? ? 3 + tan x 2 3 tan x 2 ? ? ? ? ? ? + C (2) -tan x 2 = tKsinx = 2t 1 + t2 ,tanx = 2t 1 t2 , dx = 2dt 1 + t2 u Z dx sinx + tanx = Z 1 t2 2t dt = 1 2 ln|t| t2 4 + C = 1 2 ln ? ? ?tan x 2 ? ? ? 1 4 ? tan x 2 ?2 + C (3) Z xdx 5 + x x2= Z xdx s 21 4 ? x 1 2 ?2 = Z x 1 2 s 21 4 dx ? x 1 2 ?2+ 1 2 Z dx s 21 4 ? x 1 2 ?2 = s 21 4 ? x 1 2 ?2 + 1 2 arcsin x 1 2 21 2 + C = p 5 + x x2+ 1 2 arcsin 2x 1 21+ C 113 (4) -t = 4 1 + x4Kx = 4 t4 1,dx = t3(t4 1) 3 4dt u Z 1 x 4 1 + x4dx = Z t2 t4 1 = 1 4 Z ? 1 t 1 1 t + 1 ? dt+1 2 Z 1 1 + t2 dt = 1 4 ln ? ? ? ? t 1 t + 1 ? ? ? ?+ 1 2 arctant+ C = 1 4 ln ? ? ? ? 4 1 + x4 1 4 1 + x4 + 1 ? ? ? ?+ 1 2 arctan( 4 p 1 + x4) + C (5) Z xdx 2 + 4x= 1 2 Z xd4x + 2 = 1 2x 2 + 4x 1 2 Z (2 + 4x) 1 2dx = x 2 2 + 4x 1 12(2 + 4x) 3 2+ C (6) Z cosx 1 + sinx dx = Z dsinx 1 + sinx = ln(1 + sinx) + C (7) - 6 x = tKx = t6, dx = 6t5 dt u Z dx x(1 + 2x + 3 x)= 6 Z dt t(1 + 2t3+ t2) = 6 Z ?1 t 1 4(t + 1) 6t 1 4(2t2 t + 1) ? dt q Z 6t 1 4(2t2 t + 1) dt = 3 8 Z d(2t2 t + 1) 2t2 t + 1 +1 8 Z dt 2t2 t + 1 = 3 8 ln|2t2t+1|+ 1 47 arctan 4t 1 7+ C1 l Z dx x(1 + 2x + 3 x)= 6ln|t| 3 2 ln|t+1| 9 4 ln|2t2t+1| 3 27 arctan 4t 1 7+C = 6ln| 6 x| 3 2 ln| 6 x+1|9 4 ln|2 3 x 6 x+1| 3 27 arctan 4 6 x 1 7+C = 3 4 ln x 3 x (1 + 6 x)2(2 3 x 6 x + 1)3 3 27 arctan 4 6 x 1 7+ C (8) Z x + 1 x 1 x + 1 +x 1dx = Z (x + 1 x 1)2 (x + 1 + x 1)(x + 1 x 1)dx = Z (x p x2 1)dx = x2 2 x 2 p x2 1 + 1 2 ln|x + p x2 1| + C (9) - 3 r x + 1 x 1 = tKx = t3+ 1 t3 1, dx = 6t2 (t3 1)2 dt u Z dx 3 p(x + 1)2(x 1)4= 3 2 Z dt = 3 2t + C = 3 2 3 r x + 1 x 1 + C (10) - 4 x = tKx = t4, dx = 4t3 dt u Z dx x(1 + 4 x)3= 4 Z t (1 + t)3 dt = 4 Z ? 1 (1 + t)2 1 (1 + t)3 ? dt = 4 1 + t + 2 (1 + t)2 + C = 4 1 + 4 x+ 2 (1 + 4 x)2+ C (11) Z dx ax2 + bx + c = 1 a Zd ? a ? x + b 2a ? ? a ? x + b 2a ?2 + 4ac b2 4a = 1 aln ? ? ? ? a ? x + b 2a ? + p ax2+ bx + c ? ? ? ?+ C (12) - 4 r a x x = t u Z xdx 4 px3(a x)= Z 4at2 (1 + t4)2 dt = 4a Z ? t (t2+ 2t + 1)(t2 2t + 1) ?2 dt = a 2 Z dt (t2 2t + 1)2a 2 Z dt (t2+ 2t + 1)2+a Z dt t4+ 1 q Z dt (t2 2t + 1)2= Zd ? t 2 2 ? “? t 2 2 ?2 + 1 2 #2= 2t 2 2(t2 2t + 1)+ 2arctan(2t 1) + C 1 Z dt (t2+ 2t + 1)2= 2t + 2 2(t2+ 2t + 1)+ 2arctan(2t + 1) + C 2 Z 1 t4+ 1 dt = 1 2 Z t2+ 1 t4+ 1 dt 1 2 Z t2 1 t4+ 1 dt 114 Z t2+ 1 t4+ 1 dt = Z 1 + 1 t2 t2+ 1 t2 dt = Zd ? t 1 t ? ? t 1 t ?2 + 2 = 1 2arctan t2 1 2t+ C3, Z t2 1 t4+ 1 dt = 1 22 ln t2 2t + 1 t2+ 2t + 1+ C4 l Z xdx 4 px3(a x)= at3 1 + t4 a 2 2 arctan 2t 1 t2 + a 42 ln ? ? ? ? t2+ 2t + 1 t2 2t + 1 ? ? ? ?+ a 22 arctan t2 1 2t+C t = 4 r a x x . (13) Z x p x4+ 2x2 1dx = 1 2 Z p (x2+ 1)2 2dx2= x2+ 1 4 p x4+ 2x2 11 2 ln(x2+1+ p x4+ 2x2 1)+ C (14) Z p 2 + x x2dx = Z s 9 4 ? x 1 2 ?2 dx = 2x 1 4 p 2 + x x2+ 9 8 arcsin 2x 1 3 + C (15) Z x2dx 1 + x x2= Z p 1 + x x2dx+ Z x + 1 1 + x x2dx = 2x 1 4 p 1 + x x25 8 arcsin 2x 1 5 p 1 + x x2+ 3 2 arcsin 2x 1 5+ C = 2x + 3 4 p 1 + x x2+ 7 8 arcsin 2x 1 5+ C (16) Z x2+ 1 xx4+ 1 dx = Z 1 + 1 x2 r x2+ 1 x2 dx = Zd ? x 1 x ? s? x 1 x ?2 + 2 = ln x 1 x + r x2+ 1 x2 ! +C = ln ? ? ? ? x2 1 + x4 + 1 x ? ? ? ?+ C (17) Z sin6xdx = Z ?1 cos2x 2 ?3 dx = 1 8 Z (1 3cos2x + 3cos22x cos32x)dx = 1 8x 3 16 sin2x + 3 16 Z (1+cos4x)dx 1 16 Z cos22xdsin2x = 1 8x 3 16 sin2x+ 3 16x+ 3 64 sin4x 1 16 sin2x+ 1 48 sin32x+ C = 5 16x 1 4 sin2x + 3 64 sin4x + 1 48 sin32x + C (18) Z sin2xcos4xdx = 1 5 Z sinxdcos5x = 1 5 sinxcos5x + 1 5 Z cos6x = 1 5 sinxcos5x + 1 5 ? 5 16x 1 4 sin2x + 3 64 sin4x + 1 48 sin32x ? + C = 1 16x 1 20 sin2x + 3 320 sin4x + 1 240 sin32x 1 5 sinxcos5x + C (19) Z sin4xcos4xdx = Z ?sin2x 2 ?4 dx = 1 16 Z ?1 cos4x 2 ?2 dx = 1 64 Z (1 2cos4x + cos24x)dx = 3 128x sin4x 128 + 1 1024 sin8x + C (20) Z cos4x sin3x dx = 1 2 Z cos3xd 1 sin2x = 1 2 cos3x sin2x 3 2 Z cos2x sinx dx = cos3x 2sin2x 3 2 Z dx sinx + 3 2 Z sinxdx = cos3x 2sin2x 3 2 Z sec2 x 2 tan x 2 dx 2 3 2 cosx = cos3x 2sin2x 3 2 ln ? ? ?tan x 2 ? ? ? 3 2 cosx + C (21) Z dx sin3xcos5x = Z sin2x + cos2x sin3xcos5x dx = Z dx sinxcos5x + Z dx sin3xcos3x = Z sin2x + cos2x sinxcos5x dx + Z sin2x + cos2x sin3xcos3x dx = Z sinx cos5x dx+2 Z dx sinxcos3x+ Z dx sin3xcosx = 1 4 cos4x+2 Z sin2x + cos2x sinxcos3x dx+ Z sin2x + cos2x sin3xcosx dx = 1 4 sec4x + 2 Z sinx cos3x dx + 3 Z dx sinxcosx + Z cosx sin3x dx = 1 4 sec4x + sec2x + 3 Z dtanx tanx 1 2 csc2x = 1 4 sec4x+sec2x+3ln|tanx| 1 2 csc2x+C1= 1 4 tan4x+ 3 2 tan2x 1 2 cot2x+ 3ln|tanx| + C (22) Z tanxtan(x+a)dx = Z tanx tanx + tana 1 tanxtana dx = Z tan2x + tanxtana + 1 1 1 tanxtana dx = Z 1 + tan2x 1 tanxtana dx Z dx = Z dtanx 1 tanxtana x = cotaln|1 tanxtana| x + C1= cotaln ? ? ? ? cosx cos(x + a) ? ? ? ? x + C 115 (23) Z sin5xcosxdx = 1 2 Z (sin6x + sin4x)dx = 1 12 cos6x 1 8 cos4x + C (24) Z sin2x 1 + sin2x dx = Z 1 csc2x + 1 dx = Z ? 1 csc2x 1 + csc2x ? dx = x+ Z dcotx 2 + cot2x = x+ 2 2 arctan ? 2 2 cotx ? + C (25) sin(a b) 6= 0 K Z dx sin(x + a)sin(x + b) = 1 sin(a b) Z sin(x + a) (x + b) sin(x + a)sin(x + b) dx = 1 sin(a b) Z ?cos(x + b) sin(x + b) cos(x + a) sin(x + a) ? dx = 1 sin(a b) ln ? ? ? ? sin(x + b) sin(x + a) ? ? ? ?+ C (26) I = Z xexcosxdx = xexcosx Z ex(cosxxsinx)dx = xexcosx Z excosxdx+ Z xexsinxdx = xexcosx sinx + cosx 2 ex+xexsinx Z ex(sinx+xcosx)dx = xexcosx sinx + cosx 2 ex+xexsinx sinx cosx 2 ex Z xexcosxdx + C1= ex(xcosx + xsinx sinx) I + C1 KI = Z xexcosxdx = ex 2 (xcosx + xsinx sinx) + C (27) -tanx 2 = tKsinx = 2t 1 + t2 ,cosx = 1 t2 1 + t2 , dx = 2dt 1 + t2 u Z dx (2 + cosx)sinx = Z 1 + t2 t(3 + t2) dt = Z ? 1 3t + 2t 3(3 + t2) ? dt = 1 3 ln|t(t2+3)|+C = 1 3 ln ? ? ?tan x 2 ? tan2 x 2 + 3 ? ? ?+ C = 1 6 ln (1 cosx)(2 + cosx)2 (1 + cosx)3 + C (28) Z ln(x + p 1 + x2)2dx = xln(x + p 1 + x2)2 Z x 1 (x + 1 + x2)2 2(x + p 1 + x2) ? 1 + x 1 + x2 ? dx = xln(x + p 1 + x2)2 Z d(1 + x2) 1 + x2= xln(x + p 1 + x2)2 2 p 1 + x2+ C (29) Z sinxcosx sinx + cosx dx = Z sin2 ? x + 4 ? 1 2 2sin?x + 4 ? dx = 2 2 Z sin ? x + 4 ? dx 1 22 Z dx sin ? x + 4 ? = 2 2 cos ? x + 4 ? 1 22 Z d ? tan ?x 2 + 8 ? tan ?x 2 + 8 ?= 1 2(sinx cosx) 2 4 ln ? ? ?tan ?x 2 + 8 ? ? ? + C (30) Z lnx (1 + x2) 3 2 dx = Z lnxd ? x 1 + x2 ? = xlnx 1 + x2 Z dx 1 + x2= xlnx 1 + x2 ln(x + p 1 + x2) + C (31) Z xexsinxdx = xexsinx Z ex(sinx + xcosx)dx = xexsinx Z exsinxdx Z xexcosxdx = xexsinx sinx cosx 2 ex ex 2 (xcosx + xsinx sinx) + C = ex 2 (xsinx xcosx + cosx) + C (32) Z x3arccosx 1 x2dx = x2arccosx p 1 x2+2 Z xarccosx p 1 x2dx Z x2dx = x2 p 1 x2arccosx x3 3 2 3(1 x 2)3 2arccosx 2 3 Z (1 x2)dx = x2 p 1 x2arccosx x3 3 2 3(1 x 2)3 2arccosx 2 3 ? x x3 3 ? + C = 6x + x 3 9 2 + x2 3 p 1 x2arccosx + C (33) Z (x+|x|)2dx = Z (2x2+2x|x|)dx = 2 3x 3+2 Z xsgnxxdx = 2 3x 3+2 3x 3sgnx+C =2 3x 3+2 3x 2|x|+C (34) Z x2excosxdx = x2excosx Z ex(2xcosxx2sinx)dx = x2excosxex(xcosx+xsinxsinx)+ x2exsinx Z ex(2xsinx + x2cosx)dx = x2ex(cosx + sinx) ex(xcosx + xsinx sinx) ex(xsinx xcosx + cosx) Z x2excosxdx + C1= ex(x2cosx + x2sinx 2xsinx + sinx cosx) I + C1 KI = Z x2excosxdx = ex 2 (x2cosx + x2sinx 2xsinx + sinx cosx) + C 116 (35) Z xex (1 + x)2 dx = Z xexd 1 1 + x = xex 1 + x + Z exdx = xex 1 + x + ex+ C = ex x + 1 + C (36) Z xln2 xdx = 2 3 ln2x x 3 2 4 3 Z x 1 2lnxdx = 2 3 ln2x x 3 2 8 9x 3 2lnx + 8 9 Z xdx =2 3 ln2x x 3 2 8 9x 3 2lnx + 16 27x 3 2+ C = 2 27x 3 2(9ln2x 12lnx + 8) + C (37) -t = r b x x aKx = b + at2 1 + t2 ,x a = b a 1 + t2 , dx = 2(b a)t (1 + t2)2 dt u Z dx p(x a)(b x)= 2 Z dt 1 + t2 = 2arctant + C = 2arctan r b x x a + C (38) Z xln 1 + x 1 x dx = x2 2 ln 1 + x 1 x Z x2 1 x2 dx = x2 2 ln 1 + x 1 x Z dx 1 x2 + Z dx = 1 2x 2 ln 1 + x 1 x 1 2 ln 1 + x 1 x + x + C (39) Z xarctanx ln(1 + x2)dx = x2 2 arctanxln(1 + x2) 1 2 Z x2 ? ln(1 + x2) 1 + x2 + 2xarctanx 1 + x2 ? dx = x2 2 arctanxln(1+x2)1 2 Z ln(1+x2)dx+1 2 Z ln(1 + x2) 1 + x2 dx Z xarctanxdx+ Z xarctanx 1 + x2 dx = x2 2 arctanxln(1+x2)x 2 ln(1+x2)+ Z x2 1 + x2 dx+1 2 arctanxln(1+ x2) Z xarctanx 1 + x2 dx + Z xarctanx 1 + x2 dx x2 2 arctanx + 1 2 Z x2 1 + x2 dx = x2 2 arctanxln(1 + x2) x 2 ln(1 + x2) + 3 2x 3 2 arctanx + 1 2 arctanxln(1 + x2) x2 2 arctanx + C = 1 2 arctanxx2ln(1 + x2) + ln(1 + x2) x2 3 x 2 ln(1 + x2) + 3 2x + C (40) Z sinh2xcosh2xdx = 1 4 Z sinh22xdx = 1 8 Z (cosh4x 1)dx = 1 32 sinh4x x 8 + C 117 1 1Vg |O (1) Z l 0 f(x)dxf(x) = ax + b,a,b (2) Z 2 1 x2dx (3) Z 1 0 axdx ) (1) f(x)30,lY73m0,lnKzfmxi= l n izfmxi1,xim:=i= i nl(i = 1,2, ,n). n X i=1 f(i)xi= n X i=1 (ai+ b)xi= n X i=1 ? ia n l + b ? l n = n X i=1 (nb + ia) l n2 = bl + n + 1 2n al2 u Z l 0 f(x)dx =lim |x|=1 n0 n X i=1 (ai+ b)xi= lim n ? bl + n + 1 2n al2 ? = bl + a 2 l2 (2) x231,2Y73m1,2nKzfmxi= 3 nizfmxi1,xim:=i = 1 + 3i n = 3i n n (i = 1,2, ,n). n X i=1 f(i)xi= n X i=1 2 ixi= n X i=1 ? 3i n n ?2 3 n = n X i=1 3 n3 (9i2 6ni+ n2) = 9 2 ? 1 + 1 n ? 2 + 1 n ? 9 ? 1 + 1 n ? + 3 u Z 2 1 x2dx =lim |x|=1 n0 n X i=1 2 ixi= lim n ? 9 2 ? 1 + 1 n ? 2 + 1 n ? 9 ? 1 + 1 n ? + 3 ? = 3 (3) ax30,1Y73m0,1nKzfmxi= 1 n izfmxi1,xim:=i= i n(i = 1,2, ,n). n X i=1 f(i)xi= n X i=1 aixi= n X i=1 1 na i n= a 1 n(1 a) n(1 a 1 n) ,a 6= 1 1,a = 1 u Z 1 0 axdx =lim |x|=1 n0 n X i=1 aixi= lim n a 1 n(1 a) n(1 a 1 n) = a 1 lna ,a 6= 1 1,a = 1 118 23 1. ?e5 (1) f(x)32,2k.Y: 1 n(n = 1,2,3,) (2) f(x) = sgn ? sin x ? 30,1. ) (1) f(x)32,2. f(x)32,2k.M 0|f(x)| 6 M,x 2,2l ?(f) 6 2M. 0g,NvN = ? 2M ? +1u3 ? 1 N ,2 ? f(x)kkY: f(x)3 ? 1 N ,2 ? . 3 ? 0, 1 N ? mxi1i?mxi?i(f) 6 (f) 6 2MK n X i=1 i(f)xi6 2M n X i=1 xi6 2M N 0A,B?:x 0,x00|x0 x 00| 0. yf(x) 0 “K73x0 a,bf(x0) 0 dY?530 f(x0) 2 0uk Z b a f(x)dx Z x0+ x0 f(x)dx Z x0+ x0 f(x0) 2 dx = f(x0) 0. 4. eK? (1) Z 1 0 xdx, Z 1 0 x2dx (2) Z 2 0 xdx, Z 2 0 sinxdx (3) Z 1 2 ? 1 3 ?x dx, Z 1 0 3xdx ) (1) x (0,1)x x2K Z 1 0 xdx Z 1 0 x2dx (2) x ? 0, 2 ? x sinxK Z 2 0 xdx Z 2 0 sinxdx (3) Z 1 2 ? 1 3 ?x dx = Z 1 0 ? 1 3 ?x2 dx = Z 1 0 32xdx ?x (0,1)2 x x32x 3x l Z 1 2 ? 1 3 ?x dx Z 1 0 3xdx 5. f(x)3a,bY Z b a f2(x)dx = 0yf(x)3a,b“. yy.bf(x)3a,b“Kf2(x) 0 “. 121 qf(x)3a,bYf2(x)3a,bY K13K Z b a f2(x)dx 0 Z b a f2(x)dx = 0g. ubl f(x)3a,b“. 6. f2(x)3a,b?f(x)3a,b. )f(x) = ? 1,?xkn 1,?xn f2(x) = 1 Kf2(x)3a,bYl f2(x)3a,b qd1Kf(x)3a,b. 7. f(x),g(x)3a,bYylim max(xi)0 n X i=1 f(i)g(i)xi= Z b a f(x)g(x)dxxi16 i6 xi,xi16 i6 xi(i = 1,2, ,n),xi= xi xi1(x0= a,xn= b). yf(x),g(x)3a,bYKf(x)g(x)3a,bYf(x)g(x)3a,b=lim max(xi)0 n X i=1 f(i)g(i)xi= Z b a f(x)g(x)dx lim max(xi)0 n X i=1 f(i)g(i)xi=lim max(xi)0 n X i=1 f(i)g(i)xi+ lim max(xi)0 “ n X i=1 f(i)g(i)xi n X i=1 f(i)g(i)xi # q ? ? ? ? ? n X i=1 f(i)g(i)xi n X i=1 f(i)g(i)xi ? ? ? ? ? = ? ? ? ? ? n X i=1 f(i)(g(i) g(i)xi ? ? ? ? ? 6 n X i=1 |f(i)|g(i) g(i)|xi6 n X i=1 M(f)i(g)xi= M(f) n X i=1 i(g)xiM(f)L|f|3a,b.i(g)Lg3xi