不定积分公式上课讲义

1、不定积分公式Ch4、不定积分§1、不定积分的概念与性质1、原函数与不定积分定义1:若F(x)f(x)，则称F(x)为f(x)的原函数。 连续函数一定有原函数； 若F(x)为f(x)的原函数，WJF(x)C也为f(x)的原函数；事实上，F(x)C'F'(x)f(x) f(x)的任意两个原函数仅相差一个常数。'事实上，由Fi(x)Fi(x)Fi(x)F2(x)f(x)f(x)0,得Fi(x)F?(x)C故F(x)C表示了f(x)的所有原函数，其中F(x)为f(x)的一个原函数。定义2:f(x)的所有原函数称为f(x)的不定积分，记为f(x)dx,积分号,f(x)被

2、积函数，x积分变量。显然f(x)dxF(x)CkdxkxC1xxdx12、基本积分表(共24个基本积分公式)3、不定积分的性质f(x)g(x)dxf(x)dxg(x)dxkf(x)dxkf(x)dx(k0)cscxcscxcotxdxcsc2xdxcscxcotxdxcotxcscxCdx.-2sinx22sinxcosx2cosx,22dxsinxcosx2,2cscxdxsecxdxcotxtanxC1、2、cot2xdxcsc2x1dxcotxxC第一类换元法faxbdx1、求不定积分sin5xdx君12x7dxdx,a21dx例2、求不定积分Dx.12.xdx§2、不定积分的

3、换元法（凑微分法）faxbdax1-daxbasin5xd5x5xu1sinudu=52x7d(12x)dxanx1xa2dxn,Lrctanaa一-xarcsina即xn1dxdxn1，、-cos(5x)C512x162x8C(20)(23)2x3,1x3J31x3xedx-edx-e331 1.2 cosdx1.cos-d1.1sinx2xxxxcosx,dx2cosxdx2sin、x2CC3ICdxxc1，x，x,，3、dxdlnx,edxde,sinxdxdcosx,x1dxx2dxcosxdxdsin2x,secxdxdtanx,secxtanxdxdsecx,1x12dxdarct

4、anx,1dxdarcsinx,.a2dx2xda2tanxdxdxIncosxCInsecxC(16)cosxcosxcosxdsinxcotxdxdxInsinxCIncosxC(17)sinxsinxsecxsecxtanxdsecxtanxsecxdxdxInsecxtanxC(18)secxtanxsecxtanxcscxcscxcotx,dcscxcotxcscxdxdxIncscxcotxC(19)sinxdcosx23IcotxcotxcscxcscxdInxInInxCdxxInxInxdx2cosx1dtanx1Intanxtanxtanxxe4x1edxxexeIn1dx

5、x1exex1eIn1exCxeL271edxdex2arctanexxTx2edx1x2d1x1x2例4、求不定积分2x2a2axaxa上ln2axaCxa2xx2.2x1x312xdx12xdx1112dxdx12ad(xxa)adxdx21x21dx1x2【ln2d(xa)xax23arctan(21)(22)二、sin2xdxdx5sin5xcos3xdxcotx切dxInsinxdx71sinxdx8cosxsinx2x26dx52x1lnx2cos2xdx212x2一-dx2x5x22x3一arctan2dxx1211111xcos2xd2xx-sin2xC222242sin2xd

6、x1ocos8x1-cos2xC4sin8x2cosxdxsinInsinx1sinx,2dxcosx第二类换元法1、三角代换例1、Va2x2dxdsinx16dInsinsinxlnsinxdcosx2cosxsec2xdxdx.2sinx4cscxInsinxtanxInInsinxCcosxcscx一4cot解：令xasint（或acost）.a2x2acost,dxacostdt原式=acostacostdt21cos2tadtdtcos2td2t2sin2tC2a一xarcsin2a42a42x22、t212-a2一xarcsinax.a2x2Cdx.xarcsina解：令xasin

7、tacostdtacost-x八dttCarcsinCa例3、dx解：令xatant(或acott)，贝U侦a2x22asect,dxasectdtasec2tdt原式=sectdtinsectasecttantinx2a2Inxx2a2C(24)例*4、解：令xatant(或acott),则x242sect,dx2一2sectdt原式=asec2tdtsectdtasectInsecttantCIn例5、dxHa2解：令xasect(或acsct)x2a2atant,dxasecttantdtasecttantdt原式=sectdtInsect-xtantCIn-a,x2a2c(25)inx

8、x2a2C,-x29例6、dxx解：令xasect，贝UJx293tant,dx3secttantdt2tantdt2,sect13tant.x2933arccosxCx2933arccosCx.a22xxasint小结：f(x)中含有Jx2a2可考虑用代换xatantx22axasect2、无理代换例7、dx13x1解：令3X1t,则xt31,dx3t2dt23t2dt1t311dt1t1t3t1dt3tIn1t1t2例8、x2133x13ln13x1dxx13x解：令Vxt,则xt6,dx6t5dt5日扯6t5dt原式=飞2t31t2-ydtt2dt6tarctant66xarctan6x

9、me11x_,例9、fdxt,则x解：令1xdxx,xt22tdt21原式=t22tdt"22t1t21 2dtt21产1dt,t1cInCt1例10、2.1x、xln1xxdx解：令瑚1ext,则xInt21,dx2tdtt2112t原式；-dt1dtln2ln/iZ1.一1ex14、倒代换dx例11、6xx，一.1解：令x-tt7原式t6dt124dt1ln4±ln24分部积分公式:UVUVdxd4t614t6124ln4t66x-6x§3、分部积分法UV,UVUVUVdxUVdx,故UdVUVVdU（前后相乘）（前后交换）xdsinxxsinxsinxdxx

10、sinxcosxC例2、xexdxxxxxdexee例3、lnxdxxlndxxexxxdlnexCxxlnx1xdxxln或解：令lnxt,x原式tdettettettedtteetCxxlnxxCxxC例1、xcosxdx例4、arcsinxdxxdx1x2.1x2C§4、两种典型积分1 xarcsinxxdarcsinxxarcsinxd1x2xarcsinxxarcsinx.1x2或解：令arcsinxt,xsint2原式tdsinttsintsintdttsintcostCxarcsinx,1xC例5、exsinxdxXX.xxXsinxdeesinxecosxdxesin

11、xcosxdexxesinxecosxxedcosxx_.esinxcosxexsinxdx故exsinxdx1exsinxcosxC2x例6、2dxcosxxdtanxxtanxtanxdxxtanxlnsecxC例7、lnxV1x2dx221xxlnx1xxxxlnx.1x21x21x2xdxxlnx.1xdx1x21x2C、有理函数的积分有理函数R(x)P(x)Q(x)nanxmn1anixmmbmxbmixa1xa0Kxb0可用待定系数法化为部分分式，然后积分。例1、将Xx化为部分分式，并计算%hdx解：-x3x5x6ABx3A2BA3AB12Bx3dxdx故dx565ln(x2)6l

12、n(x3)C1 x5x6x2x3或解：I2x511以1dx25x611dxx25x62x25x62x25x6I2-Inx5x621121 1dxx2211,x3八Inx5x6InC2x2例2、x1xx(x1)2dx1x(x1)1(x1)2dx2dx(x1)2Inx111例3、2x4x1dx2-dx121x2xxx121xx1arctan1xx2dx12x12x11x4124x1dx2例4、dx对三角函数有理式积分IRsinx,cosxdx,令utan-,2贝Ux2arctanu,2u1u2sinx,cosx,dx1u1u2.du，故I1u1u2Ru2,1u221u2du,三角1dx1xdx1x111x-arctanx,2<21lL2、2x1x2C2x122x122212xxx12x11,2x12xC2.2arctanln2x、.2x2x212xx2二、三角函数有理式的积分函数有理式积分即变成了有理函数积分例5、-3dx5cosx解：令uxtan-,贝Ux2arctanu2cosx1u2,dxudu原式-2.a2du1u1u35-u2duXm222x.2tanln242Cxtan2例6、dx2sinxcosx5解：令uxtan,贝Ux2arctan