不定積分習(xí)題及答案

不定積分習(xí)題及答案第四章不定積分(A層次)1．2x1dx；3．；2．；2xx2sinxcosx1x4．sin5xsin7xdx；5．a(chǎn)rctgxdx；6．x1xx1x2；7．xarctgxdx；8．2coslnxdx；9．x5x5x48x3x3x；10．x31x82dx；11．x3cosxdx；12．ex；213．；14．dxdxxlnxlnlnx1e2x；15．xexex2dx；16．sinxdx；17．3arcsinx1xdx；18．2；1xx319．7cosx3sinxxxdx；20．5cosx2sinx1x2；21．sin5xcos3xdx；3xdx；22．dx；23．dx；24．xlntgx102arccosxcosxsinx1x22arctgxdx；25．x12x12x；26．x2a2x2；27．e12；2x；e28．29．；30．xx2x2xxxxxx3。x2(B層次)1．設(shè)fx的一個(gè)原函數(shù)為sinx，求2xdx。x2．x11x；4．xlnx；3．x1xxx2dx；5．ln2x；6．lnsinxdx；7．xexdx；xsin2xe23x28．x1x322；9．1x2arcsinxdx；10．x3arccosx1x2dx；11．x33x2；12．1arctg1xdx；13．x1x21x23x；14．xx1；15．x1xx1x2；16．；217．11x1x2dx；18．dx；19．xtgxsec4xdx；acosxbsinx22220．sinx1sinxcosxdx；(C層次)1．設(shè)Fx為fx的一個(gè)原函數(shù)，F(xiàn)01，F(xiàn)x0，且當(dāng)x0時(shí)，有fxFxxex1x2，求fx。sin2x,x02．設(shè)fxx0，求fx的一個(gè)原函數(shù)。ln2x,x03．設(shè)fx為x的連續(xù)函數(shù)，且滿足方程：x0fttftxxC，求fx及常數(shù)C。12x894．設(shè)flnxln1x，計(jì)算fxdx。x25．設(shè)fx1x2x22，且fxx，求xdx。0x16．設(shè)flnx且f00，求fx。x,1x7．已知fsin2xcos2xtg2x，0x1，求fx。8．若fsinxcos12x，且f0，求方程fx0的根。23fxf2xfx9．求dx。fxfx310．maxx3,x2dx。第四章不定積分(A層次)dx1．sinxcosx解：原式2xdlnC2．2x11x2解：原式d1xdx21x1x2221x2arcsinxC3．xx2解：原式1111lnxlnx2Cx1x2331x2lnC3x14．sin5xsin7xdx解：原式1cos12xcos2xdx1cos2xdx1cos12xdx22211sin2xsin12xC4245．a(chǎn)rctgxdxx1x解：原式2x1x22xdxx2C6．x1x24解：x1x2令xt1tttttt2tt11dsintcostdt1t1lnsintcostC22sintcost2211arcsinx1x2xCln227．xarctgxdx解：原式133x133xx311x21313x3arctgx3arctgxx3arctgx3arctgx1316xdx1xdx1x32ln21x21xC68．coslnxdx解：原式xcoslnxxsinlnxdxxcoslnx1xxcoslnxxsinlnxxdsinlnxsinlnxdxxcosxlnxxsinlnxcoslnxdx故coslnxdxxcosxlnx12sinlnxC5x489．xx3x3x解：原式2xxx2xx3x813x3312x2x84dx3dxxx1x1dx11x33322x8lnx4lnxx3lnxC10．x381x2dx解：原式1dx4841x2令xu41du1sec2tdt2令utgt4sec4t1u4214costdt2181cos2tdt518t116sintC11u881u211u2C1x8C4arctgx41x811．xcos2xdx12x解：原式x21xdxxcos2xdx214x2214xdsin2x14x2214xsin2x14sin2xdx111x2xsin2x2xsin2x448cos2xC312．ex解：令3xt，則xt3，t2原式e2dt3t2etetttttdtttt2et62et6tdetetetdtt2ettet6etCe3x223x2C23xdx13．xlnxlnlnxxdlnx解：原式dxxlnlnxlnlnxC14．dx1ex2解：dx1ex2令ext1ttt1t2ttdttdtdt1t211d1t1t1tt2t1t11tC6ex1ex11exCxlnx1e11exC15．xexex2dx解：原式xde1xex2xde1x1dexexx11xexexxdxxe1ex1ex1exex1lnexlnexCxex1lnexC16．sin3xdx解：令3xt，則xt3，t2原式2dt3sinttt2dcost2cost32cost6tcosttdtttdsint2costtsint6tsintdtt2ttt6costC3x233x63x3x63xC17．a(chǎn)rcsinx1xdx解：令xsinu，則xsin2u，dx2sinucosuduu原式cosu2sinucosudu2ucosucosu2cosudu2ucosu2sinuC21xarcsinx2xC18．ln1xx327解：原式ln1x21x2d2ln21x2x212x2x21x2ln21x2x21dx2x21x22ln21x2x2112x2121x2ln1x2x2212x2ln1x2Cln21xx2x212ln21xC7cosx3sinx19．dx5cosx2sinx5cosx2sinx2cosx5sinx解：原式dx5cosx2sinx12x5x5x2xxln5cosx2sinxC20．xx1x2解：原式xxd11x11xlnxx1x1xdxxlnx1x1xdxxlnx1xlnxC21．sin5xcos3xdx解：原式sin5xcos2xcosxdx8sin5x2x1sindsinx16sin6x18sin8xC22．lntgxcosxsinxdx解：原式2dlntgxdlntgx12lntgx2C102arccosx23．dx1x2解：原式12102arccosxd2arccosx11102arcosxC2ln1012ln10102arccosxC24．x2arctgxdx解：原式13arctgxd3x1xx33311x213x31x3xx1x3211xdx1xdxxarctgx31x3332ln111xx3arctgx21xC236625．x1x2x21解：令x1，dxt1t2dt11原式t111t2t21t21tt1t1t1t2229t1t2C1xx2121xC26．x2a2ax22解：令xatgt，asec2原式ata22ta2tdt2tt2t2t2t2ttsectdtcostdtsin2tlnsecttgt1sintCa2x2xa2x2xa2axClnxa2x2a2x2C2x27．e2xsec2xtgx解：原式edxe2xsec2xdx2e2xtgxdxe2xdtgx2e2xtgxdx2xtgxtgxe2xdx2ee2xe2xtgxC28．xx2x解：原式124x2x1x310124lnx23lnx3lnxC1x24lnxx23C29．2xx解：令tgx2t，則x2，dx2dtt，sinx1t1t22，x1t1t22，于是1t原式22ttdtt2t32313t1d23t233t131t13lnt3tC1xxln3322C30．exx3xx3xx。2x解：原式xesinxcosxdxesinxtgxsecxdxxesinxdsinxesinxdsecxxdesinxsinxsecxsecxdsinxesinxesinxxesinxdxsecxesecxesinxcosxdxxxesecxC(B層次)111．設(shè)fx的一個(gè)原函數(shù)為sinxx，求2xdx解：2xdx122xd2x12x1xf2x1xdf222f2xdx12xf2x14f2xd2x由題設(shè)fxsinxxxcoxsinxx2∴2xdx12xf2x1sin2x42xCcos2xsin2x44xC2．x1x1x解：原式xedxxxex1xexxexxex1xexd11x1xxlnxexln1xexCxlnxln1xexC3．1xxx2解：原式x1x2dxx1xlnxC4．xlnxdxdx2解：原式lnx2x22lnx11x22x112x22lnx12x11x1x22lnx111x2x2x422lnxC5ln2x．x3解：原式1122x21ln2x2x2112x22lnx1x2x12x221x2ln2x2x2x12x2211x2xln2x2x2x2x214x2C6．lnsinxsin2xdx解：原式lnsinxdctgxctgxlnsinxctgxcosxsinxdxctgxlnsinxctg2xdxctgxlnsinxcsc2xdxlnsinxxCxex7．dxex2解：令ex2t，則ext故原式2lnt22dt22，xlnt22，dxtt22dt13tlnt222ttt22dt2tt24t2t222ttlnt22tarctgC

22e222xe24e242arctgCxx28．x1x322解：令ulnx,1x322，則vdvdx1x322令xtgtsec2tdtxcostdtsintCCsect1x32x取v，于是1x2x原式xd1x2xx1x2x11x2xxx1x21x2xx1x2x1x2C9．1x2arcsinxdx解：令arcsinxt，則xsint，1x2t，dxcostdt，于是原式tcos2tdt12t1cos2tdt141414t2t21414tcos2tdt11t244tsintsintdttdsint111t2tsintcost428cos2tC14arcsinx211x1x2arcsinx24x2C10．xarccosx31x2dxx3dx解：令uarccosx，dv，1x2則vdvx13dx1x22x21x22d11x2122x2d211x22x1x2211x2122xdxx21x2222d111x1x22x23211x2C取v2x32211x2，于是原式2x232x11x22x23211x2dx2x232arccos2x2x121x1x231211x2dx2x32211x2arccosx23x2dx31322x1x2arccosx19x26xC11．x33x215解：令t33x2，則x13t2，dxtdt3原式t2tt2345t2tt2451Clnt2lnt33t14513x2233x213333x213C12．11xarctgdx1x1x21x1

解：arctg1x1x2原式1x1xd1x1x11x2arctgC21x13．xx23解：原式11xd22tgx1x1dx2tgxtgx221xcsc2x1dx2tgx21xctgxxC2tgx214．xx1解：令x1t，則xt21，dxtdt16原式2lnt21dt21dttlnt212ttt12dttlnt21t1tC2t12x1lnx4x12lnx11x11C15．x1x解：令arcsinxt，則x1，tg2t，2t1x1x1x2tx，于是原式2t2tt2ttg2tsec2t1dtttg2ttgttcxxxxC1x1x1xx1xxC16．x21x2解：原式11x1x221

x1111arctgxdxarctgx2xx1x22111x2x2x1x21arctgx1arctgx1lnxln1x2C2x221717．11x1x2dx解：令xsint，則原式11tt1dt1sint1sintcos2t1dttgCcostx1x1CC1x21x21x218．dxacos2sin2x2xb解：原式1xsec2ab221tgxadx11bdtgxabba21tgxa1baC19．xtgxsec4xdx解：原式xsec3xdsecxx1x1dsec4xsec4x4442xx1sec4xtgdtgx44sec4xdxx114x3x443C20．sinx1sinxcosxdx解：xsinxdx令2t21sinxcosxsintsintcostdt1tttttCtt18xxxC222(C層次)1．設(shè)Fx為fx的一個(gè)原函數(shù)，F(xiàn)01，F(xiàn)x0，且當(dāng)x0時(shí)，有fxFxxex1x2，求fx。解：由題意Fxfx，從而FxFxxex21x2于是2FxFxdxxex1x2dx1F2xxdxdx1xx11x1dxxx1x1x1exxx1exC由F01和F201C得C0，從而Fxex1xx故fx221x32sin2x,x02．設(shè)fxx0，求fx的一個(gè)原函數(shù)。ln2x,x0解：當(dāng)x0時(shí)，有Fxfxdxsin2xdx12cos2xC1當(dāng)x0時(shí)，有Fxfxdxln2xdxxln2xx22x1dxxln2xx12ln2xC219FxCFxC1lim

2，lim12x0x0故C2C112因而，給C不同的值，便可得到fx不同原函數(shù)11cos2xC21fx011xln2xln2xxC2213．設(shè)fx為x的連續(xù)函數(shù)，且滿足方程：x0fttftxxC，求fx及常數(shù)12x89C。解：將等式兩端對x求導(dǎo)，得fxx2fx2x2x2x有fx2xx2x1x2將fx2x代入原式得代入原式得2ttxxxC10x89即x1xxxC899891故C9ln1x4．設(shè)flnx，計(jì)算fxdx。x解：令lnxt，則xe，ft1ettetfx1exexln1exdex201exln1ex1exdxeln1exxex1dx1exexln1exxln1exCx1ezln1exC25．設(shè)fx1x2x22，且fxx，求xdx。解：顯然fx2xln22x1111則fxlnx1x1又∵fxx∴x1x1x1于是xdxx1x12dxdxx2lnx1Cx1x10x16．設(shè)flnx