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题目1证明题容易。证明)()()()(afxfdttftxdxdxa解答_。)()()()()()()()()()()()()()()()(afxfxfafdttftxdxddttfafxadttfaxtftxtdftxdttftxxaxaxaxaxa题目2证明题容易。利用积分中值定理证明0sinlim:400dxxnn解答_。使上存在点在由积分中值定理0sinlim0sinlim1sin0sinlim4]4[0,()04(sinlimsinlim,]4,0[,4000040xdxdxxnnnnnnnnnnQ题目3证明题一般。使内至少存在一点证明:在,内可导,且在设函数0)(f],[0)(0)(],[)(badxxfafbaxfba解答_。使,在一点应用罗尔定理,可知存上,在区间,使存在一点由积分中值定理,在0)(b)(a,)(a,][0)(0))(()(),(11111fafabfdxxfbaba题目4证明题一般。为正整数时证明:当,设anadxxfndxxfnaxfxf00)()()()(解答_。证明:anaaaanaanaaaaaaaaaaanaanaaanadxxfndxxfdxxfdyyfdyanyfanyxdxxfdxxfdyyfdyayfdyayfayxdxxfdxxfdyyfdyayfayxdxxfaxfxfdxxfdxxfdxxfdxxf00000)1(0000320002)1(200)()()()())1(()1()()()()()2(2)()()()()()()()()()()(题目5证明题一般。证明:)1()1(1010dxxxdxxxmnnm解答_。时时且则令证10100110)1()1()()1()1(0,11,01:dxxxdtttdtttdxxxtxtxdtdxtxmnmnnmnm题目6证明题一般。且上可积在则有上任意两点且对上有定义在设2)(21)()()(,],[)(.)()(,,],[,],[)(abafabdxxfbaxfyxyfxfyxbabaxfba解答_。有由定积分的不等性质即又由题设知上可积在于是上连续在因为证明2220)(21)()()(2)()()()(2)()]()([)()]()([,)()()()()()()()(.],[)(,],[)(0lim)()(),(:abafabdxxfabafabdxxfabdxaxafdxxfdxaxafaxafxfaxafaxaxafxfbaxfbaxfyxxfxxfybaxbababababax题目7证明题一般。其中证明且内可导在上的连续在设)(sup,)()(4:.0)()(,),(,],[)(2xfMabMdxxfbfafbabaxfbxaba解答_。有两式相加有取绝对值故又由有定理由假设并利用微分中值证明2222222i2211)(4)(,)(8)()()(8)()()()()()(,2,1.)()(sup),()()()()()(),()()()()()(,:abMdxxfabMdxxbMdxxfabMdxaxMdxxfMxbxfMaxxfiMfxfMbxfbxbfxfxfxafaxafxfxfbabbabbabaabaabxa题目8证明题一般。使,内至少存在一点上正值,连续,则在在设bbdxxfdxxfdxxfbabaxfaa)(21)()(),(],[)(解答_从而原式成立。又即使在一点由根的存在性定理,存时,由于证:令aaaaaaaxa)(2)()()()()()()(0)F(b)(a,0)()(0)()(0)(],[)()()(dxxfdxxfdxxfdxxfdxxfdttfdttfdttfdttfbFdttfaFxfbaxdttfdttfxFbbbbbbbxQ题目9证明题一般。证明:sinsin020201xdxxdxnn解答_202012012020100010012010101sinsin00sinsin)sin(sin0)sin1(sinsinsin],2,0[]2,0[)sin1(sinsinsin0sin0sin]2,0[.]2,0[sinxdxxdxxdxxdxdxxxxxxxxxxxxxdxxxxnnnnnnnnnnnnnnn,由性质,有使且连续非负,在又已知函数,由性质,有,使非负,且连续在已知函数证明:题目10证明题一般。求证:10326421xxdx解答_。又时,103210210232232332326421642121414121440244)1,0(xxdxxdxdxxxxxxxxxxxxx题目11证明题一般内恒等于零。在区间上积分为零,证明内任一闭上连续,且在在区间设),()(),(),()(baxfbabaxf解答_。而从而则由题设。令,证明:设0)()()(0)(0)()()(),(),(00xfxfxxxdttfxbaxbaxxx题目12证明题一般。证明上连续在若函数0)(a)(21)(:,]1,0[)(20023aadxxxfdxxfxxf解答_。时,时,,且,则令证22200002322)(21)(2121)()(0021:aaaadxxxfdtttfdtttfdxxfxataxtxdtxdxtx题目13证明题一般。证明上连续在和设函数bababadxxgdxxfdxxgxfbaxgxf)()(])()([:,],[)()(222解答_。即所以其判别式此二次式均非负且对任意的二次三项式不等式左端是关于即故有上连续并由题设知它在显然为参数的定积分考虑以])(][)([])()([0])(][)([])()([0.,,0)()()(2)(0])()([,],[.0)]()([])()([222222222222bababababababababababadxxgdxxfdxxgxfdxxgdxxfdxxgxfttdxxfdxxgxftdxxgtdxxtgxfbaxtgxfdxxtgxft题目14证明题一般4020)dsin)(cos2(sindcos)2(sin]1,0[)(。证明:上连续,在设ffxf解答_右式。左式,,则在第二个积分中,令左式40404040400424244020)dsin)(cos2(sindsin)2(sindcos)2(sindsin)2(sinsintd)2(sind(-t))2cos())2(sin(dcos)2(sin-dtd2t-22dcos)2(sindcos)2(sindcos)2(sinffffttfttfftfff题目15证明题一般。证明且上可导在设2)(2)(:,0)(,)(,],[)(abMdxxfafMxfbaxfba解答_。有由定积分的比较定理又则微分中值定理上满足在由假设可知证明2)(2)()(,)()(),(M,(x)fx)(a,))(()()()(,],[)(),(,:abMdxaxMdxxfaxMxfbaxaxfafxfxfxaxfbaxbaba题目16证明题一般。证明:上连续,,在设aadxxafxfdxxfaaxf020)]2()([)()0(]2,0[)(解答_。,则令由于aaaaaaaadxxafxfdttafdxxfdxxfdtdxtaxdxxfdxxfdxxf000202020)]2()([)2()()(2)()()(题目17证明题一般。;为正整数,证明:设sin)2(cos)1(22kxdxkxdxk解答_。。)02()02(]2sin4121[22cos1sin)2()02()02(]2sin4121[22cos1cos)1(22kxkxdxkxkxdxkxkxdxkxkxdx题目18证明题一般。试证且上有一阶连续导数在设1)]([:.1)0()1(.]1,0[)(210dxxfffxf解答_。证明11)]0()1([2101)(2)(2)]([1)(2)]([01)(2)]([]1)([:1010102222ffxfdxdxxfdxxfxfxfxfxfxf题目19证明题一般。证明:为正整数,若2020cos21sincosxdxxdxxmmmmm解答_右式。左式,则令cos21cos21cos21)2(sin21)2(21)2(sin21sincos20202212212020xdxtdttdtdtttxdxxxdxxmmmmmmmmmmmm题目20证明题一般。则上连续,在区间若函数])([)()(],[)(babadxxabafabdxxfbaxf解答_。时时且则作代换101010])([1])([1)]()([)(01)(,)(dxxabafabdttabafabdtabtabafdxxftaxtbxdtabdxtabaxba题目21证明题一般。证明:上连续在设函数2020)cos(41)cos(,]1,0[)(dxxfdxxfxf解答_。得证则令在后一积分中为周期的函数是以显然证2020202020202020022220020)cos(41)cos()cos(4])cos()cos([2)cos()cos()cos())cos(()cos(,])cos()cos([2)cos(2)cos()cos(:dxxfdxxfdxxfdxxfdxxfdxxfdxxfdttftdtfdxxftxdxxfdxxfdxxfdxxfxf题目22证明题一般。,则连续,且在若函数0)()()()(xfdttfxfRxfxa解答_。已知常数考虑函数有且可导在连续在RxxfcceafdttfafceexfccxpexfxfexfexfxpRxexfxpxfxfxfdttfxfRxRxfRxfxaaxxxxxxxa0)(00)(0)()(f(x))()()(0)]()([)()()(.)()(0)()()())(()()()(1题目23证明题一般。证明:为周期的连续函数,是以设)()2()()(sin)(020dxxfxdxxfxxxf解答_。,则令证明:由于
本文标题:定积分的证明题
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