数学分析数学分析数学分析 (7).pdf

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1、?l?0?F?adxx)(?F?adxxf)(?,)a?)()(0 xKxf?K?F?adxx)(?F?adxxf)(?F?adxxf)(F?adxx)(?Cauchy?F?adxx)(?0?/1?aA.000,AAA./?KdxxAA1?F)(?FAAdxxf)(1?FAAdxxK)(?F?adxxf)(?Cauchy?F?adxxf)(00?01?aA./00,AAA.0?1KdxxfAA.F)(?.FAAdxx)(?0)(11.FAAdxxfK?F?adxx)(?,)a?0)(,0)(.xxf?0)()(lim?xxfx?F?F?F?adxxf)(?adxx)(?F?adxxf)(?ad

2、xx)(?278?21)(xxf?)20(1)(?pxxp?0)()(lim?xxfx?F?1)(dxxf?F?1)(dxx?21?p?10?p?,)a?0)(,0)(.xxf?)()(limxxfx?F?F?F?adxxf)(?adxx)(?F?adxxf)(?adxx)(?xxf1)(?)21(1)(?pxxp?)()(limxxfx?F?1)(dxxf?F?1)(dxx?121?p?1?p?Cauchy?,)a?(,)0?f x().0K?f xKxp()?p?1F?adxxf)(?f xKxp().?p?1F?adxxf)(?,)a?(,)0?f x().0lim()xpx f xl?

3、0?l?p?1F?adxxf)(279?0?l?p?1?F?adxxf)(?)(x?px1?11321xexdxx?Fln;?F?131tanarcdxxx;?110?Fxxdx|sin|;?xxdxqp11?F?.?Rqp,?1?x?1ln123?xexx?231x?11321xexdxx?Fln?2?x31arctanxx?32x?F?131tanarcdxxx?3?0.xxxx?.?11sin11?dxxF?011?110?Fxxdx|sin|?4?xpqxx?1?qpx?1?280?1?qpxxdxqp11?F?xxdxqp11?F?f x()cpv(f x dx()?Ff x dx(

4、)?F?f x dx()?F)cpv(f x dx()?F0)(.xf)cpv(f x dx()?Ff x dx()?F?)cpv(f x dx()?F?Alim?)(AF?AlimF?AAdxxf)(?Cauchy?0?/1?00A0?0,AAA./?1?)()(AFAF?0,AAA./0,ABB./?FAAdxxf)(1?)()(AFAF?F?BBdxxf)(1?)()(BFBF?F?0)(dxxfF?0)(dxxff x dx()?F?lnlnlnsinxxxdx2?F;?sin xxdxp1?F?;?Rp?F?1tanarcsindxxxxp?;?Rp?sin()xdx20?F;?F?

5、anmxdxxqxpsin)()(?q?n?pxm()xn()mqx()n?),?ax?1?F?AxdxAF2sin)(xxlnlnln?),2?0lnlnlnlim?xxx?Dirichlet?lnlnlnsinxxxdx2?F?281?.xxxsinlnlnlnxxx2sinlnlnln)2cos1(lnlnln21xxx?F?2lnlnlndxxx?F?22coslnlnlnxdxxx?F?2sinlnlnlndxxxx?lnlnlnsinxxxdx2?F?2?1?pppxxx1sin?F?11dxxp?1?psin xxdxp1?F?10?pF?AxdxAF1sin)(px1?),1?

6、01lim?pxx?Dirichlet?sin xxdxp1?F?10?pF?1|sin|dxxxp?10?p?sin xxdxp1?F?3?1?p?pxxxarctansinpx2?F?11dxxp?1?pF?1tanarcsindxxxxp?10?pF?AxdxAF1sin)(pxxarctan?),1?0arctanlim?pxxx?Dirichlet?F?1arctansindxxxxp?10?pF?1sinarctandxxxxp?10?p?F?1arctansindxxxxp?4?2xt?F?02)sin(dxxF?02sindttt?F?02sindttt?sin()xdx20?

7、F 282?5?1?mnxxxqxpnmsin)()(2xK?1?mnF?anmxdxxqxpsin)()(?1?mnF?AxdxAF1sin)(x)()(xqxpnm?0)()(lim?xqxpnmx?Dirichlet?F?anmxdxxqxpsin)()(?x?)()(xqxpnm?xa?F?1sin)()(dxxxqxpnm?1?mnF?anmxdxxqxpsin)()(?1?mnAxqxpnmx?)()(lim?A?F?anmxdxxqxpsin)()(?f x(),a bxb?3?Cauchy?,?)a bf x().0 x?b,b?B)b0K?f xKbxp()()?p?1?f

8、x dxab()F?f xKbxp()().?p.1f x dxab()F?p?1F?bapdxxb)(1?Cauchy?2830?/1?0?03?),0(,3BB?/?Kdxxbbbp1BB?F?)(1?F?)(BBbbdxxf1BB?F?)(bbpdxxbK?f x dxab()F?1.pF?bapdxxb)(1?Cauchy?00?01?0?/3?),0(,3BB?0?Kdxxbbbp0)(11BB.?F?.F?)(BBbbdxxf0)(1BB.?F?bbpdxxbK?f x dxab()F?Cauchy?,?)a bf x().0lim()()xbpbxf xl?0?l?p?1?f x

9、 dxab()F?0?l?p.1f x dxab()F?lim()()xbpbxf xl?lp0,1?0?03?),(bbx3?/?pxblxf)(1)(?lim()()xbpbxf xl?.lp0,1?0?03?),(bbx3?/?pxblxf)(2)(?f x g x dxab()()F 284?Abel?,?f x dxab()Fg x()a b?Dirichlet?F?BBbadxxfF)()(,0(ab?g x?(),)a b0)(lim?xgbx?1?Cauchy?Gxg?|)(|f x dxab()F0?/1?0?03?),(,bbAA3?/?GdxxfAA2)(1?F?FAAd

10、xxgxf)()(FF2?2?AAdxxfAgdxxfAg55)()()()(?FF?AAdxxfGdxxfG55)()(111?22?2?MF?|)(|B?),baAA?/?MdxxfAA2)(?F?0)(lim?xgbx0?/1?0?03?),(bbx3?/?Mxg4)(1?FAAdxxgxf)()(FF2?2?AAdxxfAgdxxfAg55)()()()(|)(|2|)(|2AgMAgM?111?22?Cauchy?F?adxxgxf)()(?112301xxdx()?F;?ln xxdx2011?F;?12202cossinxxdx?F;?102?Fcosxxdxp?;?|ln|xd

11、xp01F;?xxdpq?F11011()x;?F?1011|ln|)1(dxxxxqp.?1?32)1(1xx?321x)0(?x?32)1(1xx?31)1(1x?)1(?x?285?112301xxdx()?F?2?1lnlim21?xxx21?10?3?01lnlim20?xxxx3?0?x3xxx11ln2?ln xxdx2011?F?3?xx22sincos1?21x)0(?x?xx22sincos1?2)2(1x?)2(?x?12202cossinxxdx?F?4?pxxcos1?221?px)0(?x?3?p?102?Fcosxxdxp?3.p102?Fcosxxdxp?5?1

12、0?3?p0|ln|lim0?pxxx30?x3xxp1ln?pxln?px?)1(1)1(?x?1?p|ln|xdxp01F1?p?|ln|xdxp01F?6?11)1(?qpxxpx?11)0(?x?11)1(?qpxxqx?1)1(1)1(?x?0,0?qpxxdpq?F11011()xxxxdpq?F11011()?7?|ln|)1(11xxxqp?qx?)1(1)1(?x?0|)ln|)1(lim11210?xxxxqppx?0?x21111ln)1(pqpxxxx?1,0?qp?F?1011|ln|)1(dxxxxqpF?1011|ln|)1(dxxxxqp 286?xxxdxpq

13、?F1101ln?;?Rqp,?112230 x xxdx()()?F;?ln()10?Fxxdxp;?F?0tanarcdxxxp?;?F2/0tan?dxxxp;?xdpx?F10ex;?10 xxdxpq?F;?F?2ln1dxxxqp.?1?xxxdxpq?F1101lnF?2101lndxxxpF?2101lndxxxqF?12111lndxxxxqp?0?p0?qF?2101lndxxxp?F?2101lndxxxq?1x?xxxqpln11?*?*?)1(1ln1)1(11)1(111?xxxqp?qpxxqp?1)1)(?F?12111lndxxxxqp?xxxdxpq?F11

14、01ln?2?F?032)2()1(1dxxxxF?1032)2()1(1dxxxxF?2132)2()1(1dxxxxF?232)2()1(1dxxxx?32)2()1(1?xxx?313121x2?)0(?x?32)2()1(1?xxx?32)1(1?x)1(?x?F?1032)2()1(1dxxxx?287?32)2()1(1?xxx?32)1(1?x)1(?x?32)2()1(1?xxx?313)2(121?2x)2(?x?F?2132)2()1(1dxxxx?32)2()1(1?xxx?313)2(121?2x)2(?x?32)2()1(1?xxx?341x)(?x?F?232)2(

15、)1(1dxxxx?112230 x xxdx()()?F?3?F?0)1ln(dxxxp?F10)1ln(dxxxpF?1)1ln(dxxxp?pxx)1ln(?11?px)0(?x?2?p?F?10)1ln(dxxxp?2.pF?10)1ln(dxxxp?1?p0)1ln(lim213?!#+,-?2?ppxxxx?0?x2131)1ln(?ppxxx?1213?p?1?pF?1)1ln(dxxxp?1?pF?1)1ln(dxxxp?288?21?pF?0)1ln(dxxxp?F?0)1ln(dxxxp?4?F?0tanarcdxxxpF?10tanarcdxxxpF?1tanarcdxx

16、xp?pxxarctan?11?px)0(?x,?2?p?F10tanarcdxxxp?pxxarctan?px2?)(?x?1?pF?1tanarcdxxxp?21?pF?0tanarcdxxxp?F?0tanarcdxxxp?5?F2/0tan?dxxxpF?4/0tan?dxxxpF?2/4/tan?dxxxp?pxxtan?211?px)0(?x?23?p?F4/0tan?dxxxp?23.p?F4/0tan?dxxxp?pxxtan?122()2ppx?)2(?x?F2/4/tan?dxxxp?23?p?F2/0tan?dxxxp?23.p?F2/0tan?dxxxp?6?xdxF?

17、101edxxxppx?F10eF?11edxxxp?F?11edxxxpxpex?1px?11)0(?x?0?pxdpx?F10ex0?p?xdpx?F10ex 289?7?10 xxdxpq?FF?101dxxxqpF?11dxxxqp?qp?10 xxdxpq?F?qp?qpxx?1?),min(1qpx)0(?x?qpxx?1?),max(1qpx)(?x?1),min(?qp?1),max(?qp10 xxdxpq?F?10 xxdxpq?F?8?1?pqx211ln1?pqpxxx?121?p?F?2ln1dxxxqp?1?pqx211ln1?pqpxxx?121?p?F?2ln1

18、dxxxqp?1?ptx?lnF?2ln1dxxxqpF?2lnqtdt?1?p1,1?qpF?2ln1dxxxqp?F?2ln1dxxxqp?xxdxp?F1201;?xxxdxqpsin11?F?;p.0?F?0sincosedxxxpx;?F?0sin2sinedxxxpx;290(5)F1021cos1dxxxp?(6)F?()$%&?11sindxxxxp?.?0?p?1?xxdxp?F1201F?10211dxxxpF?1211dxxxp?211xxp?px?11)0(?x?211xxp?px?31)(?x?20?pxxdxp?F1201?xxdxp?F1201?2?1?pqqpp

19、qxxxx?11|sin|?xxxdxqpsin11?F?pqp?1?F?AxdxAF1sin)(xpqxx?1?01lim?pqxxx?Dirichlet?F?11sindxxxxpq?F?11|sin|dxxxxpq?pqp?1?sin xxdxp1?F?pq.?n?22sin1qnpnxxdxx?F?xxxdxqpsin11?F?3?F?0sincosedxxxpxF?10sincosedxxxpxF?1sincosedxxxpx?pxxxecossin?px1)0(?x?1?p?F10sincosedxxxpx?F10sincosedxxxpx?1?pF?1sin|cos|edxxxp

20、x?0?p?291F?1sincosedxxxpx?10?p1cos1sin?FexdxeAx?px1?01lim?pxx?Dirichlet?F?1sincosedxxxpx?10?pF?0sincosedxxxpx?F?0sincosedxxxpx?4?F?0sin2sinedxxxpxF?10sin2sinedxxxpxF?1sin2sinedxxxpx?pxxxe2sinsin?12?px)0(?x?2?p?F10sin2sinedxxxpx?F10sin2sinedxxxpx?21?pF?1sin|2sin|edxxxpx?1?pF?1sin|2sin|edxxxpx?0?p?F?1

21、sin2sinedxxxpx?10?pF?)1(sin02sinkkxxdxeFAxxdxe0sin2sin?px1?01lim?pxx?Dirichlet?F?1sin2sinedxxxpx?21?pF?0sin2sinedxxxpx?10?pF?0sin2sinedxxxpx?F?0sin2sinedxxxpx?292?5?21xt?F?1021cos1dxxxptdttpcos121123F?1?p?F1021cos1dxxxp?31?pF1021cos1dxxxp?3.pF1021cos1dxxxp?6?1?pppxxxx11sin?()$%&?F?()$%&?11sindxxxxp?

22、10?pF?()$%&?261sin?nnpdxxxxpn()$%&?2?2321?)$&?11npn?(%2?F?()$%&?11sindxxxxp?F?dxxxxp1)1sin(dxxxxxxpF?1sin1coscos1sin?xpxx1sin?pxx1cos?Dirichlet?F?()$%&?11sindxxxxp?F?()$%&?11sindxxxxp?10?F?04sinsinxdxxx 293?AAA?F?4sinsinAAxdxxxF?42)(cos4sinAAxdxx244cossinAAxxx()$%&?F?244coscosAAdxxxxF342sincosAAdxxx

23、x?Cauchy?AF?04sinsinxdxxx11?f x()x?0?f x()?f x dx()01Flim()xxf x?00?f x()10?x?F?xxdttfxfx2)()(20?Cauchy?F?xxxdttf200)(lim?0)(lim0?xxfx?12?F?adxxf)()(xxf),?a?0)()(lnlim?xfxxx?x?)(xxf00)(.xxf?F?2?xxdttttfxfxx1)()()(ln210Fxxdttf)(?Cauchy?0)(lim?F?xxxdttf?0)()(lnlim?xfxxx?13?f x()lim()xf x?0?fx()?,?)0?F

24、fxx dx()sin20 294?F?02sin)(xdxxfF?02)(sinxxdfF?02sin)(xdxxf?F?AxdxAF02sin)(f x()lim()xf x?0Dirichlet?F?02sin)(xdxxf?Ffxx dx()sin20?F?adxxf)(lim()xf x?0?fx dxa2()?F?lim()xf x?0aA?0?Ax?/?1)(?xf?Ax?)()(2xfxf?F?adxxf)(?fx dxa2()?F15?,fx dxa2()?Ff x()a?,?a b?f x()f x()?F?adxxf)(fx dxa2()?Fxxxfsin)(?F?1)(

25、dxxfF?12)(dxxffx dxa2()?F?F?adxxf)(xxf1)(?F?12)(dxxf?F?1)(dxxf 295?fx dxa2()?FF?adxxf)(xxxfsin)(?F?12)(dxxfF?1)(dxxfF?adxxf)(?fx dxa2()?F?01,)(23?nnnnxnxf?F?1)(dxxfF?12)(dxxf?)(1 21)(2xfxf?Fbadxxf)(2Fbadxxf)(?Fbadxxf)(Fbadxxf)(2xxf1)(?F10)(dxxf?F102)(dxxf?sinsinxxxdxp?F1?p?12?121?p?p?1?p?1xxxxpsinsin?px2?F?12dxxp?sinsinxxxdxp?F1?10?p)sin(sinsinsinsin2xxxxxxxxxpppp?F?1sindxxxp?F?12)sin(sindxxxxxpp?121?p?210?p?121?p?F?434sinsin?nnpdxxxx1)1(122?2.ppn?1)1(11?ppnn?F?1sinsindxxxxp?296?121?p?sinsinxxxdxp?F1?210?p?sinsinxxxdxp?F1?0?pF?2242sinsin?nnpdxxxx2224sin2nnxdx?F?162?Cauchy?sinsinxxxdxp?F1?297