基本积分公式表

基本积分公式表kdx)1(kxC)(是常数kdxx)2(11x,C)1(xdx)3(||lnxCdxx211)4(Cxarctandxx211)5(Cxarcsinxdxcos)6(Cxsinxdxsin)7(Cxcosxdx2sec)8(Cxtanxdx2csc)9(Cxcotr5t6yuux总裁的天价小妻子xdxxtansec)10(Cxsecxdxxcotcsc)11(Cxcscdxex)12(Cexdxax)13(Caaxln第二节换元积分法（一）一、第一换元积分法问题dxex2?,2数不是积分公式表上的函被积函数xe用直接积分法，求不出它的积分。怎么办？dxex2xe2)2(xd2121xe2)2(xdxu221uedu21ueC21xe2C一般情况下：,)()(uFuf有原函数设)()('ufuF即)(duufuF)(C可导若)(xu)(uF)]([xF)]([xFdxd)(')('xuF)(uf)('x)]([xf)('x的原函数是)(')]([)]([xxfxFdxxxf)(')]([)]([xFC)(uFCduuf)(:,我们就得到下面的定理这样设)(uf具有原函数，)(xu可导，则有换元公式定理1)(xudxxxf)(')]([duuf)(:用法dxxg)()]([xf)('xdx)]([xf)]([xd)(ufdu)(xu)(uFC)]([xFC)(xu是积分公式表上的函数)(uf“凑”微分法例1求.2sinxdx解1xdx2sinx2sin)2(xd21x2sin)2(xd21xu2令usindu21)cos(u21C)]2cos([x21C)2cos(x21C解2xdx2sinxxcossin2dxxsin)(sinxd2xusin令udu222u2C2uCx2sinC解3xdx2sinxxcossin2dxxcos)(cosxd2xucos令udu222u2C2uCx2cosCxxsincos2dx)1(xcos)(cosxd2例2dxx231x231)23(xdxu23令u1du21||lnu21C|23|ln21xC21x231)23(xd21例3dxx)43(sec2)43(sec2x)43(xd43xu令u2secdu31utan31C)43tan(31xC31)43(sec2x)43(xd31例4dxxx2121x)1(2xd21xu令udu21212121x)1(2xd212332uCCu2331232)1(31xC例5xdxtanxxcossindxxcos1)(cosxdxucos令u1du)1(xcos1)(cosxd||lnuCCx|cos|ln类似可得xdxcotCx|sin|ln例6)0(122adxxa12a(221ax)dx121a221axdx121a2)(1ax)(axda1a12)(1ax)(axdaxarctana1C例7)0(122adxxa)1(1222axadx1a1221axdx1a12)(1ax)(axda12)(1ax)(axdaxarcsinC例8)0(122adxax))((1axaxdxa21)11(axaxdxa21)11(axaxdxa21ax1dxa21ax1dxa21ax1)(axda21ax1)(axd||lnaxa21a21||lnaxC||lnaxaxa21C例9dxxx)ln21(1xln211)ln21(xd|ln21|ln21xC21xln211)ln21(xd21例10dxxex3xe3)3(xd32xe332xe3)3(xd32C例11sin3xdxxxsinsin2dx)(cosxd)cos1(2x)cos31(cos3xxCCxx3cos31cos例12cossin52xdxxx2sindx22)sin1(x)(sinxdx2sinx2sin)sinsin21(42xx)(sinxd)sinsin2(sin642xxx)(sinxd3sin3xx5sin527sin7xCxxcoscos4例13cos2xdxdxdx)22cos21(xdx21dxx22cos2xdxx2cos212x)2(2cosxxd4122cos1xCxx42sin2类似可求cos4xdx例14cos4xdxdxdx)42cos22cos41(2xx2)22cos1(xdx)24cos14122cos41(xxdx)84cos22cos83(xxdx83dxx4cos81dxx2cos21x83)4(4cosxdx)2(2cosxdx41321Cxxx4sin3212sin4183例15cscxdxdxdx2cos2sin21xxxsin1dx2cos2cos2sin212xxxdx2cos2tan212xxdx2sec2tan212xx)2(xd2sec2tan12xx)2(tanxd2tan1x|2tan|lnxC2tanx2cos2sinxx2cos2sinxx2sin22sin2xxxsin2sin22xxsinxcos1xxsincosxsin1xcotxcscxdxcscCx|2tan|lnCxx|cotcsc|ln例16xdxsecdxxcos1dxx)2sin(1)2()2sin(1xdx例15Cxx|)2cot()2csc(|lnCxx|tansec|ln例17xdx6secxdxx24secsec)(dx4secxtan22)tan1(x)(tanxd)tantan21(42xx)(tanxdxtanx3tan325tan5xC例18xdxx35sectanxdxxxxtansecsectan24)(d22)1(secxxsecxxx224sec)1sec2(sec)(secxd)secsec2(sec246xxx)(secxdx7sec71x5sec52x3sec31Cx2sec例19xdxx2cos3cos)cos5(cos21xxdx21dxxx)cos5(cos(21xdx5cosxdxcos)51[21)5(5cosxxdxdxcos]51(21x5sinxsin)101x5sinxsin21CC例20.)1(3dxxx])1(1)1(1[32xxC)1(xd1)1(x2)1(21xCx112)1(21xdxxx3)1(11])1(1)1(1[32xxdx例21求.11dxex解dxex11dxeeexxx11dxeexx11dxeedxxx1)1(11xxededx.)1ln(Cexx例22求.)11(12dxexxx解,1112xxxdxexxx12)11()1(1xxdexx.1Cexx例23求.12321dxxx原式dxxxxxxx123212321232dxxdxx12413241)12(1281)32(3281xdxxdx.12121321212323Cxx例24求解.cos11dxxdxxcos11dxxxxcos1cos1cos1dxxx2cos1cos1dxxx2sincos1)(sinsin1sin122xdxdxx.sin1cotCxxdxxxx)sincossin1(22例24求另解.cos11dxxdxxcos11dxx2cos212)2(2cos12xdx)2(2sec2xdxCx2tan解例25设求.,cos)(sin22xxf)(xf令xu2sin,1cos2uxuuf1)(duuuf1)(Cuu221Cxxxf221)(例26求解.2arcsin412dxxxdxxx2arcsin412dxxx2arcsin21212)2(2arcsin2112xdxx)2(arcsin2arcsin1xdxCx|2arcsin|lnTheendofPart1作业：P2072(2)-(34)(双)