首页 高等数学积分表推导全过程

高等数学积分表推导全过程

高等数学积分表推导全过程高等数学积分表推导过程(一)含有ax+b的积分udx111u1u(ax+b)1.=d(ax+b)=lnax+b+C2.(ax+b)dx=(ax+b)d(ax+b)=+C∫ax+ba∫ax+ba∫a∫a(u+1)x1ax1ax+b−b1ax+bbd(ax+b)13.dx=dx=dx=dax−=(ax+b−blnax+b)+C∫ax+ba∫ax+ba∫ax+ba2∫ax+ba2∫ax+ba222222x1ax1(ax+b)−2abx−b12d(ax+b)4.dx=dx=dx=(ax+b)d(ax+b)−2bd(ax...

高等数学积分表推导过程(一)含有ax+b的积分udx111u1u(ax+b)1.=d(ax+b)=lnax+b+C2.(ax+b)dx=(ax+b)d(ax+b)=+C∫ax+ba∫ax+ba∫a∫a(u+1)x1ax1ax+b−b1ax+bbd(ax+b)13.dx=dx=dx=dax−=(ax+b−blnax+b)+C∫ax+ba∫ax+ba∫ax+ba2∫ax+ba2∫ax+ba222222x1ax1(ax+b)−2abx−b12d(ax+b)4.dx=dx=dx=(ax+b)d(ax+b)−2bd(ax+b)−b∫ax+ba2∫ax+ba2∫ax+ba3∫∫∫ax+b11=(ax+b)2−2b(ax+b)−b2lnax+b+Ca32dx−a111111ax+b5.=−dx=−(lnax+b−lnax)+C=−lnax+b+lna+C+lnx=−ln+C∫x(ax+b)b∫ax+baxb1bb1bxdx11a2a11a2dxadx1aa6.=+−=+−=−++−+∫2∫2dx∫2dx2∫2∫2lnaxb2lnxCx(ax+b)bxb(ax+b)bxbxbax+bbxbxbb1aax+b=+ln+Cbxb2xxdx1b1dxbdx1b17.=−=−=++⋅+∫2∫2dx∫∫22lnaxb2C(ax+b)a(ax+b)a(ax+b)aax+ba(ax+b)aaax+b1b=lnax+b++Ca2ax+b2122bxb2(ax+b)−−22x2x2bxdxbdx1b8.=aaa=−−=−+−+∫2dx∫2dx2∫22∫23ax2blnaxbC(ax+b)(ax+b)aa(ax+b)a(ax+b)aax+bdxadxadx1dx11lnx11ax+b9.=−−+=−lnax+b++C=−ln+C∫x(ax+b)2∫b(ax+b)2b2∫ax+bb2∫xb(ax+b)b2b2b(ax+b)b2x（二）含有ax+b的积分12310.ax+bdx=ax+bd(ax+b)=(ax+b)+C∫a∫3a1b252b32311.xax+bdx=(ax+b)ax+bdx−ax+bdx=(ax+b)−(ax+b)+C=(3ax−2b)(ax+b)+C∫a∫a∫5a23a215a22122bb272b212.x2ax+bdx=(ax+b)ax+bdx−xax+bdx−ax+bdx=(ax+b)−[(3ax−2b)⋅∫a2∫a∫a2∫7a3a15a2232b323(ax+b)]−(ax+b)+C=(15a2x2−12abx+8b2)(ax+b)+C3a3105a3１高等数学积分表推导过程xdx1ax+bbdx1bd(ax+b)232b13.=dx−=ax+bdx−=(ax+b)−ax+b+C=∫ax+ba∫ax+ba∫ax+ba∫a2∫ax+b3a2a22=(ax−2b)ax+b+C3a2x21(ax+b)22bxdxb2dx132b14.dx=dx−−=(ax+b)2d(ax+b)−ax+bd(ax+b)+∫ax+ba2∫ax+ba∫ax+ba2∫ax+ba3∫a3∫b2dx2=(3a2x2−4bx+8b2)ax+b+Ca2∫ax+b15a3dx15.当b>0时，有∫xax+bdxa111ax+b−b=−dax+b=ln+C∫xax+b∫2bax+bax+b−bax+b+bbax+b+bt−b1当b<0时，令ax+b=t,则dx=d=dtaa1dtdx2121t2t=a==−=−−⋅=+∫∫∫dt2∫b2dt∫b2darctanCxax+bt−bt−bb−b−b−bttta1+1+bb2ax+b=arctan+C−b−b1ax+b−bln+C(b>0)dxb所以∫=ax+b+bxax+b1ax+barctan+C(b<0)−b−bax+2babx−bax+bdxax+2badx+adx+a16.=dx−=2axbdx−=−2axbdx−∫x2ax+b∫2bx2ax+b2b∫xax+b∫bx22b∫xax+b∫b2x22b∫dxu′v−uv′adxax+badx=−dx−=−−xax+b∫v22b∫xax+bbx2b∫xax+bax+babadxdx17.dx=(+)dx=dx+b=2ax+b+b∫x∫ax+bxax+b∫ax+b∫xax+b∫xax+bax+bab2dxdxax+b1dx18.dx=(+)dx=b+a=−b−b⋅a∫x2∫xax+bxax+b∫x2ax+b∫xax+bbx2b∫xax+b２高等数学积分表推导过程dxax+badx+a=−+∫xax+bx2∫xax+b（三）含有x2±a2的积分dxππ19.设x=atant(−1时有∫22n(x+a)dxxx2x1a2=−+2(n−1)dx=−+2(n−1)−−dx∫22n22n1∫22n22n1∫22n122n(x+a)(x+a)(x+a)(x+a)(x+a)(x+a)x21x即I−=+2(n−1)(I−−aI)于是I=+(2n−3)I−n122n−1n1nn2n−1n1(x+a)2a(n−1)(x2+a2)1xdx12n−3dx由此作递推公式并由I=arctan+C即可得I∴=+1n∫22n222n−12∫22n−1aa(x+a)2(n−1)a(x+a)2(n−1)a(x+a)dx111111x−a21.=⋅dx=−dx=ln=C∫x2−a2∫x+ax−a2a∫x−ax+a2ax+a(四)含有ax2+b(a>0)的积分adxdx1dx1a1a22.==b=+(>)∫2∫∫2arctanxCb0ax+bba2bbabb1+xab1+xbdxdx1111ax−−b==−dx=ln+C(b<0)∫ax2+b∫(ax+−b)(ax−−b)2−ab∫ax−−bax+−b2−abax+−bxdx1dx2123.==lnax2+b+C∫ax2+b2∫ax2+b211ax1a11x224.dx=−dx=lnx−⋅lnax2+b+C=ln+C∫2∫22(ax+b)xbxb(ax+b)bb2a2bax+bx2dx1ax2+bb1xbdx25.=dx−dx=−∫ax2+ba∫ax2+ba∫ax2+baa∫ax2+bdxdxadx1adx26.=−=−−+C∫x2(ax2+b)∫bx2b∫ax2+bbxb∫ax2+b３高等数学积分表推导过程dx1a2xa1aa211aa27.=+−=−−+⋅2++=−−2+∫32∫3222dx22lnx2lnaxbC22lnx2lnx(ax+b)bxb(ax+b)bx2bxbb2a2bx2b2b2aax+b1ax2+b+C=ln−+C2b2x22bx2dxb−ax211b−ax21dx1u′(ax2+b)−2axu1dx28.=+dx=dx+=dx+∫22∫222∫22∫2∫22∫2(ax+b)2b(ax+b)2b(ax+b)2b(ax+b)2bax+b2b(ax+b)2bax+bau′x2+bu′−2axu=b−ax2u′b=bu′=1u=xdxx′(ax2+b)−2ax⋅x1dx1x1dx=dx+=⋅+∫22∫22∫22∫2(ax+b)(ax+b)2bax+b2bax+b2bax+b(五)含有ax2+bx+c(a>0)的积分−21dxab2−4ac29.=+−当b2<4ac时，有∫2∫axdxax+bx+c2b4ab−14a22dx2ax+dxab−4ac22a=ax+−dx=4ac−b令t=则∫2∫∫22ax+bx+c2b4ab4ac−b2ax+2a+14ac−b2b2ax+2a4a4ac−b2dt122adt=dx则原式=⋅=arctant+C=arctan+C22∫22224ac−b4ac−b2a1+t4ac−b4ac−b4ac−bb4a2ax+−22dx2a2a当b>4ac时有=4ac−bdx令t=则dt=dx∫2∫222ax+bx+c2bb−4acb−4ac4ax+2a1−b2−4ac4ab2−4acdt12ax+b−b2−4ac则原式==ln+C2∫222b−4ac2a1−tb−4ac2ax+b+b−4ac22ax+barctan+C(b2<4ac)22dx4ac−b4ac−b综上所述=∫22ax+bx+c12ax+b−b−4ac2ln+C(b>4ac)22b−4ac2ax+b+b−4ac４高等数学积分表推导过程xdx2ax+bbdx12ax+bbdx130.=dx−=dx−=lnax2+bx+c∫ax2+bx+c∫2a(ax2+bx+c)∫2a(ax2+bx+c)2a∫ax2+bx+c2a∫ax2+bx+c2a2′bdx′(ax+bx+c)2ax+b−其中[ln(ax2+bx+c)]==2a∫ax2+bx+c(ax2+bx+c)ax2+bx+c（六）含有x2+a2(a>0)的积分dxππ31.由于1+tan2t=sec2t，不妨设x=atant−0=ln++C1=ln(x+x+a)+C∫22aax+aaadxππ32.设x=atant−0)的积分dxπ45.当x>a时，设x=asect0a=−=−ln(u+u−a)+C1==−ln(−x+x−a)∫22∫22x−au−a1−x−x2−a222+C1=ln+C1=ln2+C1=ln(−x−x−a)+Cx2−a2−xadx综上所述，=lnx+x2−a2+C∫22x−a７高等数学积分表推导过程dxπ346.，设x=asect00时有∫22xx−a2dxasecttantta=dt=+C=arccos+C∫22∫xx−aasecttantaxdx1adx1a当x<0时有，=arccos+C，综上所述，有=arccos+C∫22∫22xx−aa−xxx−aaxdxπ52.，设x=asect00时，有∫x2９高等数学积分表推导过程x2−a2atantadx=asecttantdt=atan2tdt=a(sec2t−1)dt=atant−t+C=x2−a2−arccos+C∫x∫asect∫∫xx2−a2ax2−a2a当x<0时，有dx=(−x)2−a2−arccos+C；综上所述，dx=x2−a2−arccos+C∫x−x∫xxx2−a2π58.dx，设x=asect00)的积分dx1dxx59.==arcsin+C∫22∫2a−xaxa1−adxππ360.,令x=asint−0)的积分dx1dxb73.=，令x+=t，则dx=dt∫2∫2ax+bx+cabcb22ax++−2aa4a2b2−4acdx1dx1dt当b2−4ac>0时，则令=u2(u>0)，则==2∫2∫2∫224aax+bx+cabcb2at−ux++−2aa4a2再令t=usecr,dt=utanrsecrdr,t2−u2=utanr,于是1dt1utanrsecr11=dr=secrdr=lnsecr+tanr+C∫22∫∫1at−uautanraabx+222t2ax+b1−cosrt−u1secr==2a=;tanr===2aax2+bx+cub2−4acb2−4accosrub2−4ac4a2１２高等数学积分表推导过程dx12ax+b+2aax2+bx+c1=ln+C=ln2ax+b+2aax2+bx+c+C∫221ax+bx+cab−4aca4ac−b2bdx1dx1dt当b2−4ac<0时，则令=u，t=x+，==∫2∫2∫222a2aax+bx+cabcb2at+ux++−2aa4a21dt1usec2rdr11令t=utanr，dt=usec2rdr==secrdr=lnsecr+tanr+C∫22∫∫1at+uausecraa4ac−b2b2x+1u24ac−b1t2ax+b==4a=；tanr==2a⋅2a=222222secrt+ub4ac−b2a2bcu4ac−b4ac−bx++x+x+2a4a2aadx12aax2+bx+c2ax+b1∴=ln++C=ln2ax+b+2aax2+bx+c+C∫2221ax+bx+ca4ac−b4ac−badx1综上所述，=ln2ax+b+2aax2+bx+c+C∫2ax+bx+ca2b4ac−b2bb2−4ac74.ax2+bx+cdx=ax++dx当b2−4ac>0时，令t=x+，=u∫∫2a4a22a2a∫ax2+bx+cdx=a∫t2−u2dt，再令t=usecr，dt=usecrtanrdr1at2−u2dt=a⋅u2tan2rsecrdr=a⋅u2(secrtanr+lnsecr+tanr)−a⋅u2lnsecrtanr+C∫∫22111b2−4ac2ax+b1=a⋅u2secrtanr−a⋅u2lnsecr+tanr+C=a⋅2aax2+bx+c⋅−1222222ab−4acb−4acab2−4ac2ax+b+2aax2+bx+c2ax+b4ac−b2222ln+C1=ax+bx+c+ln2ax+b+2aax+bx+c+C24ab2−4ac4a8a3b4ac−b2当b2−4ac<0时，令t=x+，=u；ax2+bx+cdx=at2+u2dt；再令t=utanr，2a2a∫∫dt=usec2rdr，于是1a4ac−b2bc2aax2+bx+cdx=au2sec3rdr=a⋅u2(secrtanr+lnsecr+tanr)=⋅x2+x+⋅×∫∫22224aaa4ac−b１３高等数学积分表推导过程2ax+b14ac−b22aax2+bx+c2ax+b2ax+b4ac−b22+a⋅2ln++C1=ax+bx+c+ln4ac−b224a4ac−b24ac−b24a8a32ax+b+2aax2+bx+c+C2ax+b4ac−b2综上所述，ax2+bx+cdx=ax2+bx+c+ln2ax+b+2aax2+bx+c+C∫34a8axdxxdxb4ac−b275.=；令t=x+，当∆>0时，令u=∫2∫22ax+bx+cb4ac−b22a4aax++2a4a2bt−22xdx11tbdtt−ub=2adt=dt−=−lnt+t2−u2+C∫2∫22∫22∫221ax+bx+cat−uat−u2aat−ua2aa1bcbbbcax2+bx+cb=x2+x+−lnx++x2+x++C=−ln2ax+b+2aax2+bx+c+Caaa2aa2aaa1a2aab2t−4ac−bxdxtdtb当∆<0时，令u=，于是=2adt=−∫2∫22∫222aax+bx+cat+uat+u2aabcx2+x+dt1bbbbc=t2+u2−lnt+t2+u2+C=aa−lnx++x2+x++C∫2211t+ua2aaa2aa2aaaax2+bx+c1=−ln2ax+b+ax2+bx+c+Ca2aaxdxax2+bx+cb综上所述，=−ln2ax+b+ax2+bx+c+C∫2ax+bx+ca2aadx1dxb4ac+b276.=；令t=x−，u=，于是∫2∫22c+bx−axab2−4acb2a4a−x−4a22adx1dt1u12ax−b==−arcsin+C=arcsin+C∫2∫222c+bx−axau−tatab+4ac2b2+4acbbb2+4ac77.c+bx−ax2dx=a−x−；令x−=t；u=；于是∫∫4a22a2a4a2１４高等数学积分表推导过程tu2t1bcb1c+bx−ax2dx=au2−t2dt=u2−t2+arcsin+C=x−+x−x2⋅+∫∫2a2au22aaaab2+4ac2ax−b2ax−bb2+4ac2ax−barcsin+C=c+bx−ax2+arcsin+C8aab2+4ac4a8aab2+4acxdx1xdxb4ac+b278.=；令t=x−；u=；于是∫2∫22c+bx−axab2+4acb2a4a−x−4a22abt+xdx11tb11b2ax−b=2adt=dt+dt=−u2−t2+arcsin+C∫2∫22∫22∫222c+bx−axau−tau−t2aau−ta2aab+4ac1cbb2ax−b1b2ax−b=−+x−x2+arcsin+C=−c+bx−x2+arcsin+Caaa2aab2+4aca2aab2+4acx−a（十）含有±或(x−a)(b−x)的积分x−bx−ax−abt−aa−b79.dx；令t=；则x=；dx=dt；于是∫x−bx−bt−1(t−1)2x−aa−bt1t1dx=tdt=(a−b)dt=(b−a)td=(b−a)+(a−b)dt∫x−b∫(t−1)2∫(t−1)2∫t−1t−1∫t−1ta−b11tb−ad(t+1)b−ad(t−1)t=(b−a)+−dt=(b−a)+−=(b−a)t−12∫t−1t+1t−12∫t+12∫t−1t−1b−at+1x−ax−ab−a2x−a+ln+C=(b−a)⋅+ln(b−a)(x−a+x−b)+C=(x−b)+(b−a)ln(x−a2t−11x−bx−b21x−bx−a+x−b)+C=(x−b)+(b−a)ln(x−a+x−b)+C1x−bx−ax−abt+ab−a80.dx；令=t；则x=；dx=dt；于是∫b−xb−x1+t(1+t)2x−at1tdttdx=(b−a)dt=(a−b)td=(a−b)−(a−b)=(a−b)−(a−b)arcsint+C∫b−x∫(1+t)2∫1+t1+t∫1+t1+t１５高等数学积分表推导过程x−ax−a=(b−x)+(b−a)arctan+Cb−xb−xdxdxdtdt81.;令x−a=t；则x=a+t；dx=dt；于是==∫∫∫2∫(x−a)(b−x)(x−a)(b−x)t(b−a−t)−t+(b−a)tdtb−aa−b=；令=u>0；r=t+则∫2(a−b)2a−b22−t+42dtdr2x−a−b==arcsin+C∫2∫22(a−b)2a−bu−rb−a−t+422(a−b)2a−b82.(x−a)(b−x)dx；令x−a=t；则x=a+t；dx=dt；于是(x−a)(b−x)dx=−t+dt∫∫∫42b−aa−bru2r2x−a−b令=u>0；r=t+；则(x−a)(b−x)dx=u2−r2dr=u2−r2+arcsin+C=22∫∫22u4(b−a)22x−a−b(x−a)(b−x)dx+arcsin+C8b−a(十一)含有三角函数的积分sinx83.sinxdx=−cosx+C84.cosxdx=sinx+C85.tanxdx=dx=−lncosx+C∫∫∫∫cosxcosx86.cotxdx=dx=lnsinx+C∫∫sinxπxπxπdx+d+dtan+dx22424xπ87.∫secxdx=∫=∫=∫=∫=lntan++Ccosxππxπ2xπxπ242sinx+cosx+tan+cos+tan+22242424x2xxsin2sin1−cosxππtan=2=2==cscx−cotx;secxdx=lncscx+−cotx++C=x∫2cossinxsinx222lnsecx+tanx+Cxdxdxdxdtanx88.cscxdx====2=lntan+C=lncscx−cotx+C∫∫∫xx∫xx∫xsinx2sincostancos2tan22222289.∫sec2xdx=tanx+C90.∫csc2xdx=−cotx+C91.∫secxtanxdx=secx+C92.∫cscxcotxdx=−cscx+C１６高等数学积分表推导过程1−cos2xx11+cos2xx193.sin2xdx=dx=−sin2x+C94.cos2xdx=dx=+sin2x+C∫∫224∫∫22495.∫sinnxdx=−∫sinn−1xdcosx=−cosxsinn−1x+∫cosxdsinn−1xdx=−cosxsinn−1x+(n−1)∫sinn−2xcos2xdx=−cosxsinn−1x+(n−1)∫(1−sin2x)sinn−2xdx=−cosxsinn−1x+(n−1)∫sinn−2xdx−(n−1)∫sinnxdx1−n−1−∴sinnxdx=−cosxsinn1x+sinn2xdx∫nn∫96.∫cosnxdx=∫cosn−1xdsinx=sinxcosn−1x−∫sinxdcosn−1x=sinxcosn−1x+(n−1)∫sin2xcosn−2xdx=sinxcosn−1x+(n−1)∫(1−cos2x)cosn−2xdx=sinxcosn−1x+(n−1)∫cosn−2xdx−(n−1)∫cosnxdx1−n−1−∴cosnxdx=sinxcosn1x+cosn2xdx∫nn∫dx−1−−1−−n+1−−97.=sinnxdx=sinn1xdx=sinn1xcosx+sinn2xdx∫sinnx∫n∫nn∫dx1cosxn−1dxdx1cosxn−2dx=+∴=−+∫sinn−2xn−2sinn−1n−2∫sinnx∫sinnxn−1sinn−1xn−1∫sinn−2xdx−−−−−−−−−−−98.=cosnxdx=cosn1xdsinx=cosn1xsinx−sinxdcosn1x=cosn1xsinx−(n+1)cosn2xsin2xdx=∫cosnx∫∫∫∫−−−−dx1−−n+1−−cosn1xsinx−(n+1)cosn2xdx+(n+1)cosnxdx∴==−cosn1xsinx+cosn2xdx∫∫∫cosnxnn∫dx1sinxn−1dxdx1sinxn−1dx将n换成n−2有=−+∴=−+∫cosn−2xn−2cosn−1xn−2∫cosnx∫cosnxn−1cosn−1xn−2∫cosn−2x−1−+1+−1+−99.cosmxsinnxdx=cosm1xsinnxdsinx=cosm1xdsinn1x=sinn1xcosm1x−sinn1xdcosm1x∫∫n+1∫n+1n+1∫1−+m−1+−1−+m−1−=cosm1xsinn1x+sinn2xcosm2xdx=cosm1xsinn1x+sinnxcosm2x(1−cos2x)dxn+1n+1∫n+1n+1∫1−+m−1−m−1=cosm1xsinn1x+sinnxcosm2xdx−sinnxcosnxdxn+1m+1∫n+1∫m−11−+m−1−∴1+cosmxsinnxdx=cosm1xsinn1x+cosm2xsinnxdxn+1∫n+1n+1∫1−+m−1−∴cosmxsinnxdx=cosm1xsinn1x+cosm2xsinnxdx∫m+1m+n∫1−+1−+1+又有cosmxsinnxdx=−cosmxsinnxdcosx=−sinn1xdcosm1x=−sinn1xcosm1x+cosm1x∫∫m+1∫m+1m+1∫−1−+n−1+−1−+n−1−dsinn1x=−sinn1xcosm1x+cosm2xsinn2xdx=−sinn1xcosm1x+cosmxsinn2x(1−sin2x)m+1m+1∫m+1m+1∫1−+n−1−n−1dx=−sinn1xcosm1x+cosmxsinn2xdx−cosmxsinnxdxm+1m+1∫m+1∫１７高等数学积分表推导过程n−11−+n−1−1−+n−1∴1+cosmxsinnxdx=−sinn1xcosm1x+cosmxsinn2xdx=−sinn1xcosm1x+×m+1∫m+1m+1∫m+nm+1∫cosmxsinn−2xdx；因此1−+m−1−1+−n−1−cosmxsinnxdx=cosm1xsinn1x+cosm2xsinnxdx=−cosm1xsinn1x+cosm2xsinnxdx∫m+nm+n∫m+nm+n∫111100.sinaxcosbxdx=[sin(a−b)x+sin(a+b)x]dx=−cos(a−b)x−cos(a+b)x+C∫2∫2(a−b)2(a+b)111101.sinaxsinbxdx=[cos(a+b)x−cos(a−b)x]dx=sin(a+b)x−sin(a−b)x+C∫2∫2(a+b)2(a−b)111102.cosaxcosbxdx=[cos(a+b)x+cos(a−b)x]dx=sin(a+b)x+sin(a−b)x+C∫2∫2(a+b)2(a−b)xxax103.1+tan2dx1+tan2⋅1+tan2dxdxdx22a2−b2a2===2dx=222∫+∫xx∫2xx∫−∫absinxa+2sincosa+atan+2btanxbabxbatan+atan+22222a2a+1+1a2−b2a2−b2xbatan+2xx2asecd22d2a2a1x2aa−b22=22==⋅a−b22∫222∫2dtan22∫2a−bxba−bxb2a−baxbatan+atan+atan+2a2a2a1+1+1+a2−b2a2−b2a2−b2xbatan+22a22=arctan+C(a>b)a2−b2a2−b2104.2xa2x2ax1+tan1+tandtandxdx2b2−a2222==dx=dx=b−a2∫∫xx∫xx∫∫a+bsinx++2+xbxba2bsincosaatan2btanatan+atan+22222a2a−1−1b2−a2b2−a2x2ab2−a2atan+btantsectdt222sect22令2=sect；原式=b−aa=dt=csctdt=22∫222∫22∫22b−atantb−atantb−ab−ax+−2−22x1atanbbalncsct−cott+C=lntan+C=ln2+C(a2b)a+ba−ba−b2xxx+22dxdx1tansecdx2dtan106.∫=∫=∫2dx=∫2=∫2a+bcosx2x2x2x2xa+bb−a2xa+bcos−sin(a+b)+(a−b)tan(a+b)(b−a)tan1−tan2222a+b2b−axπxa+b令tan=sinu00）1xxxxxxx113.arcsindx=xarcsin−xdarcsin=xarcsin−xadx=xarcsin−dx=xarcsin∫∫∫2∫22aaaaxaa−xa1−a2+a2−x2+Cx1xx2x1xx2x1x2dxx2x114.xarcsindx=arcsindx2=arcsin−x2darcsin=arcsin−=arcsin+∫∫∫∫22a2a2a2a2a2a−x2a222x22axxaxx22a−x−arcsin+C=−arcsin+a−x+C44a24a4x1xx3x1x3dx115.x2arcsindx=arcsindx3=arcsin−;令x=asinu(0

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高等数学积分表推导全过程

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