常用积分公式.pdf

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1、常用积分公式（一）含有axb+的积分(a≠0)dx11．∫＝lnaxb++Caxb+aμ1μ+12．∫()ax+bdx＝()axb++C（μ≠−1）a(1μ+)x13．dx＝(laxbb+−+naxb)+C∫2axb+a2x11⎡⎤224．dx＝()ax+b−+2baxb()l++bnaxb+C∫3⎢⎥axb+a⎣⎦2dx1axb+5．∫＝−ln+Cx()axb+bxdx1aaxb+6．＝−++lnC∫22x()axb+bxbxx1b7．dx＝(lnaxb++)+C∫22()axb+aax+b22x1b8．dx＝(2axb+−+blnaxb−)+C

2、∫23()axb+aax+bdx11ax+b9．＝−+lnC∫22x()axb+baxb()+bx（二）含有axb+的积分2310．∫ax+bxd＝()axb++C3a2311．xax+bxd＝(3ax−2)(baxb++)C∫215a122222312．xax+bxd＝(15ax−+12abx8b)(axb+)+C∫3105ax213．dx＝(2ax−+b)axb+C∫2axb+3a2x222214．dx＝(3ax−4abx++8)baxb+C∫3axb+15a⎧1ax+−bb⎪ln+Cb(>0)dx⎪bax++bb15．∫＝⎨xaxb+⎪+2a

3、xb⎪arctan+Cb(<0)⎩−b−bdxax+badx16．∫2＝−−∫xaxb+bx2bxaxb+ax+bdx17．∫dx＝2axb++b∫xxaxb+ax+bax+badx18．dx＝−+∫2∫xx2xax+b22（三）含有x±a的积分dx1x19．=arctan+C∫22x+aaadxxn23−dx20．=+∫22n2221nn−−2∫221()x+a2(naxa−+1)()2(naxa−+1)()dx1xa−21．=ln+C∫22x−a2axa+2（四）含有ax+>ba(0)的积分2⎧1a⎪arctanxC+>(b0)dx⎪abb22

4、．＝⎨∫2ax+b⎪1ax−−bln+Cb(<0)⎪⎩2−+abax−bx1223．dx＝lnax++bC∫2ax+b2a2xxbxd24．dx＝−∫2∫2ax+baaaxb+2dx1x25．＝ln+C∫22x()ax+b2bax+bdx1dax26．＝−−∫22∫2x()ax+bbxbax+b2dxaax+b127．＝ln−+C∫32222x()ax+b22bxbxdxx1dx28．＝+∫2222∫()ax+b2(bax++b)2baxb2（五）含有ax++bxc(0a>)的积分⎧22axb+2arctan+

5、−−acb29．＝⎨∫22ax++bxc⎪12axb+−b−4ac2ln+>Cba(4c)⎪ba22−+42caxbba+−4c⎩x1d2bx30．dx＝lnax++−bxc∫2∫2ax++bxc22aaax+bx+c22（六）含有x+a(0a>)的积分dxx2231．＝arsh+C＝ln(x+++xaC)∫22a1x+a3dxx32．∫＝+C223222()xa+axa+x2233．∫dx＝x+aC+22xa+x134．∫dx＝−+C22322()xa+xa+22xxa222235．∫dx＝x+−axln(+++xa)Cxa22+222xx223

6、6．∫dx＝−++++ln(xxaC)22322()xa+xa+22dx1xaa+−37．∫＝ln+Cxxa22+ax22dxxa+38．＝−+C∫222ax2xxa+222xa222239．∫x+axd＝x++axln(+++xa)C22223x2222432240．∫()x+axd＝(2x+5)axaaxxa++ln(+++)C8822122341．∫xxa+dx＝()x++aC34222xa22222242．∫xxa+dx＝(2x+axa)+−ln(xxaC+++)882222xa+22xaa+−43．∫dx＝x+aa++lnCxx2222x

7、a+xa+2244．dx＝−++++ln(xxaC)∫2xx422（七）含有x−a(0a>)的积分dxxx2245．＝arch+C=lnx+−+xaC∫22xa1x−adxx46．∫＝−+C223222()x−aaxa−x2247．∫dx＝x−aC+22xa−x148．∫dx＝−+C22322()xa−xa−22xxa222249．∫dx＝x−+axln+−+xaCxa22−222xx2250．∫dx＝−++−+lnxxaC22322()xa−xa−dx1a51．∫＝arccos+Cxxa22−ax22dxxa−52．＝+C∫222ax2xxa−2

8、22xa222253．∫x−axd＝x−−axln+−+xaC22223x2222432254．∫()x−axd＝(2x−