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5p Œ˘61 SK) SSSKKK11.11. Oe 2´¨' æ 5;e´æ, .(1)R+∞0xe−xdx=(−xe−x−e−x)|+∞0=1.(2)R+∞0dx(x+1)(x+2)=lnx+1x+2|+∞0=ln2.(3)1σ√2πR+∞−∞e−(x−a)22σ2dxx−a=√2σt=1√πR+∞−∞e−t2dt=1(σ0).(4)R+∞01+x21+x4dxu=x−x−1=R+∞−∞duu2+2=1√2arctanu|+∞−∞=π√2.(5)R+∞0xsinxdx=(−xcosx+sinx)|+∞0,u .(6)R+∞4dxx√x−1t=√x−1=R+∞√32dtt2+1=2arctant|+∞√3=π3.(8)R+∞−∞dxx2+2x+2dx=arctan(x+1)|+∞−∞=π.(9)R+∞0e−xcosxdx=12e−x(sinx−cosx)|+∞0=12.(10)R1−1dx√1−x2x=sint=Rπ2−π2dt=π.(11)R120dxxlnx=lnlnx|120,u .(12)R120dxxln2x=−1lnx|120=1ln2.3. e ¨' æ 5.(1)R+∞1dxx2+3√x4+3.ˇǑ1x2+3√x4+31x2, ¨'R+∞1dxx2´æ, ¨'´æ.(2)R+∞2dx2x+3√x2+1+6.ˇǑlimx→+∞12x+3√x2+1+6/1x=12, ¨'R+∞2dxxu , ¨'u .(3)R20dx3√x+34√x+x3.ˇǑlimx→013√x+34√x+x3/14√x=13, ¨'R20dx4√x´æ, ¨'´æ.(4)R10dx3√1−x4.ˇǑlimx→113√1−x4/13√1−x=limx→113√1+x+x2+x3=13√4, ¨'R10dx3√1−x´æ, ¨'´æ.(5)R10sinxx3/2dx.ˇǑlimx→0sinxx3/2/1x1/2=1, ¨'R10dxx1/2´æ, ¨'´æ.(6)R+∞0sinxxdx.limx→0sinxx=1,¨' :.ˇǑ x→+∞ ,…Œ1x N“u0, |RA0sinxdx|≤2, )| X O{, ¨'´æ.(7)R+∞1xαe−x2dx(α0).ˇǑlimx→+∞xαe−x2e−x22=0, ¨'R+∞1e−x22dx´æ, ¨'´æ.(8)R10lnx1−xdx.limx→1lnx1−x=limx→11x−1=−1,1 · :.ˇǑlimx→0lnx1−x/lnx=1, ¨'R10lnxdx=(xlnx−x)|10=−1´æ, ¨'´æ.(9)Rπ20dxsin2xcos2x.0 π2· :.ˇǑlimx→01sin2xcos2x/1x2=1, ¨'Rπ20dxx2u , ¨'u .14. e ¨'· Ø´æ ·^ ´æ.(1)R+∞0√xcosxx+3dx.). nǑ K Œ ,R(n+1)πnπ√x|cosx|x+3dx≥R(n+1)πnπ√nπ|cosx|(n+1)π+3dx=2√nπ(n+1)π+3.Rnπ0√x|cosx|x+3dx≥n−1Pk=02√kπ(k+1)π+3.⁄–¨'RA0√x|cosx|x+3dx ., ¨' Ø´æ.ˇǑ(√xx+3)′=3−x2√x(x+3)2,⁄– x¿' ,…Œ√xx+3 N4~.dux→+∞ ,√xx+3→0,qˇǑ|RA0cosxdx|≤2,d)| X O{, ¨'´æ.n ⁄ª, ¨'^ ´æ.(2)R+∞1cos(3x+2)√x3+13√x2+1dx.).ˇǑ|cos(3x+2)√x3+13√x2+1|≤1x32x23, ¨'R+∞1dxx32x23´æ, ¨' Ø´æ.5.Qª’u ¨' )| X O{9C O{.(i))| X O{: …Œf(x),g(x)3(a,b] k‰´, a·§ :. 3~ŒM0, Ø 0εb−a,|Rba+εf(x)dx|≤M.q …Œg(x)3x→a+0 N“u0,K ¨'Rbaf(x)g(x)dx´æ.(ii)C O{: …Œf(x),g(x)3(a,b] k‰´, a·§ :.e ¨'Rbaf(x)dx´æ, …Œg(x)3(a,b] Nk.,K ¨'Rbaf(x)g(x)dx´æ.SSSKKK11.21.e …Œ 4 .(1)limk→0Rπ20dϕ√1−k2sin2ϕ.–ϕ,kǑC …Œ1√1−k2sin2ϕ3[0,π2]×[−12,12] ºY,ˇd “=Rπ20limk→01√1−k2sin2ϕdϕ=Rπ20dϕ=π2.(2)limk→1−0Rπ20p1−k2sin2ϕdϕ.–ϕ,kǑC …Œp1−k2sin2ϕ3[0,π2]×[0,1] ºY,ˇd “=Rπ20limk→1−0p1−k2sin2ϕdϕ=Rπ20cosϕdϕ=1.(4)limα→0R1+α0dx1+x2+α2. …Œ11+x2+α23 †¡ ºY,⁄–limα→0R10dx1+x2+α2=R10dx1+x2=π4.qˇǑ|R1+α1dx1+x2+α2|≤|R1+α1dx|=|α|,⁄–limα→0R1+α1dx1+x2+α2=0.n ⁄ª, “=π4+0=π4.(5)limy→0R10exsinxyy+1dx.–x,yǑC …Œexsinxyy+13[0,1]×[−12,12] ºY,ˇd “=R10limy→0exsinxyy+1dx=R100dx=0.2.e …Œ …Œ.(1)g(y)=Ra+kya−kyf(x)dx, ¥f(x)3(−∞,+∞) ºY.).g′(y)=kf(a+ky)+kf(a−ky).(2)g(y)=Rcosysinyey√1−x2dx,0≤y≤π2.2).g′(y)=−sinyeysinx−cosyeycosy+Rcosysiny√1−x2ey√1−x2dx.(3)g(y)=Ry0ln(1+xy)xdx,0y+∞.).g′(y)=ln(1+y2)y+Ry011+xydx=ln(1+y2)y+ln(1+xy)y|y0=2ln(1+y2)y.(4)g(y)=Ry20sin(x2+y2)dx,−∞y+∞.).g′(y)=2ysin(y4+y2)+Ry202ycos(x2+y2)dx.3.|^¨' e Œ {e ¨'.(1)g(a)=Rπ20arctan(atanx)tanxdx,−∞a+∞.). (x,a)→(0,a0) ,arctan(atanx)tanx=a·arctan(atanx)atanx→a0. (x,a)→(π2,a0) ,dY%‰n,arctan(atanx)tanx→0.ˇd ¿‰´ ,–x,aǑC …Œarctan(atanx)tanx3[0,π2]×R ºY.⁄–g′(a)=Rπ20∂∂aarctan(atanx)tanxdx=Rπ20dx1+a2tan2x. a0 a6=1 ,g′(a)t=tanx=R+∞0dt(1+a2t2)(1+t2)=11−a2(arctant−aarctanat)|+∞0=π2(a+1).5¿ g′(a)·??ºY …Œ, g′(a)=π2(|a|+1).qˇǑg(0)=0,¨' g(a)=π2sgna·ln(|a|+1).(2)g(a)=Rπ20ln1+acosx1−acosx·dxcosx,−1a1.). (x,a)→(π2,a0) ,acosx→0,ln1+acosx1−acosx·1acosx→2. ¿‰´ ,–x,aǑC …Œln1+acosx1−acosx·1cosx3[0,π2]×(−1,1) ºY.⁄–g′(a)=Rπ202dx1−a2cos2xt=tanx=R+∞02dt1−a2+t2=π√1−a2.ˇǑg(0)=0,⁄–g(a)=πarcsina.(3)I(a)=Rπ20ln(a2sin2x+cos2x)dx,a6=0.). …Œln(a2sin2x+cos2x)3a6=0 ºY,⁄–I′(a)=Rπ202asin2xdxa2sin2x+cos2x. a0 a6=1 ,I′(a)t=tanx=2aR+∞0t2dt(1+a2t2)(1+t2)=2aa2−1(arctant−1aarctanat)|+∞0=πa+1.5¿ I′(a)3a6=0 ºY, “I′(a)=πa+13a=1?‰⁄Æ.qˇǑI(1)=0,⁄– a0 I(a)=πlna+12.qˇǑI(a)· …Œ,⁄–I(a)=πln|a|+12.SSSKKK11.31.? e ¨'3 ‰«m ´æ5.(1)R+∞0sintx1+x2dx,(−∞t+∞).).|sintx1+x2|≤11+x2, ¨'R+∞0dx1+x2´æ, M- O{, ¨' ´æ.(2)R+∞0e−t2x2dx,(0t0t+∞).). tt00 ,|e−t2x2|≤e−t20x2,¨'R+∞0e−t20x2dx´æ, M- O{, ¨'3«m(t0,+∞) ´æ.(3)R+∞0e−αxsinxdx,(i)(0α0≤α≤+∞),(ii)(0α≤+∞).3).(i) x∈[0,+∞) 0α0≤α≤+∞ ,|e−αxsinx|≤e−α0x,¨'R+∞0e−α0xdx´æ, M- O{, ¨'3«m[α0,+∞) ´æ.(ii) α0 ,R+∞Ae−αxsinxdx=−e−αx(αsinx+cosx)1+α2|+∞A=e−αA(αsinA+cosA)1+α2.K Mıo , α=1M9A=2kπ MA2M ,|R+∞Ae−αxsinxdx|e−122.⁄– ¨'3«m(0,+∞) ´æ.(4)R+∞1e−bxcosx√xdx,(0≤b+∞).).e−bx·x N…Œ, |e−bx|≤1.¨'R+∞1cosx√xdx´æ.dC O{, ¨'3«m[0,+∞) ´æ.(5)R+∞0te−txdx,(i)(0c≤t≤d),(ii)(0t≤d).).(i) x∈[0,+∞) 0c≤t≤d ,|te−tx|≤de−cx,¨'R+∞0de−ctdx´æ, M- O{, ¨'3«m[c,d] ´æ.(ii) t0 ,R+∞Ate−txdx=−e−tx|+∞A=e−tA.K Mıo , t=1M,A=2M ,R+∞Ate−txdx=e−12.⁄– ¨'3«m(0,d] ´æ.(6)R10dxxt,(0t≤b1).). x∈(0,1] 0t≤b1 ,|1xt|≤1xb,¨'R10dxxb´æ, M- O{, ¨'3«m(0,b] ´æ.2.e ¨' .(1)R+∞0e−ax−e−bxxdx,(0ab).).R+∞0e−ax−e−bxxdx=R+∞0dxRbae−txdt.ˇǑ¨'R+∞0e−txdx3«m[a,b] ´æ,⁄– “=RbadtR+∞0e−txdx=Rba1tdt=lnba.(2)R10xa−xblnxdx,(a−1,b−1).). 5,b ab.R10xa−xblnxdx=R10dxRabxtdt.ˇǑ¨'R10xtdx3«m[b,a] ´æ,⁄– “=RabdtR10xtdx=Rab11+tdt=ln1+a1+b.(3)R+∞−14e−(2x2+x+1)dx.). t=√2(x+14), “=R+∞0e−t2−78dt√2=√π2√2e−78.(5)R+∞0sinαx·cosβxxdx(α0,β0).). “=R+∞0sin(α+β)x+sin(α−β)x2xdx=π4(1+sgn(α−β)).3.¨'R+∞0e−xsintxxdx …ŒL “.). ¨'P I(t). t∈[−a,a] x∈(0,+∞) ,|e−xsintxx|≤|e−xt|≤ae−x,¨'R+∞0ae−xdx´æ,⁄–¨'I(t)3«m[−a,a] ´æ.⁄– t∈(−a,a) ,I′(