常用积分公式

常 用 积 分 公 式

（一）含有ax?b的积分(a?0) 1．

dx1＝?ax?balnax?b?C

2．(ax?b)dx＝

??1(ax?b)??1?C（???1）

a(??1)3．

x1dx(ax?b?blnax?b)?C ＝?ax?ba2x21?1?dx＝3?(ax?b)2?2b(ax?b)?b2lnax?b??C 4．?ax?ba?2?5．

dx1ax?b＝??x(ax?b)blnx?C

6．

?dx1aax?b＝??ln?C 22x(ax?b)bxbx7．

1bx(lnax?b?)?C dx＝?(ax?b)2a2ax?b1b2x2)?C 8．?dx＝3(ax?b?2blnax?b?aax?b(ax?b)29．

?dx11ax?b＝?ln?C

x(ax?b)2b(ax?b)b2x（二）含有ax?b的积分

23(ax?b)?C ?3a2(3ax?2b)(ax?b)3?C 11．?xax?bdx＝215a22(15a2x2?12abx?8b2)(ax?b)3?C 12．?xax?bdx＝3105a10．

ax?bdx＝13．

?2xdx＝2(ax?2b)ax?b?C

3aax?b1

14．

?2x2(3a2x2?4abx?8b2)ax?b?C dx＝315aax?b??dx?15．?＝?xax?b???16．

1lnbax?b?b?C(b?0)ax?b?b

2ax?barctan?C(b?0)?b?b?x2dxax?badx＝? ??bx2bxax?bax?b17．

??dxax?b dx＝2ax?b?b?xxax?b18．

ax?bax?badx＝ dx??x2x2?xax?b22（三）含有x?a的积分 19．

dx1xarctan?C =?x2?a2aa20．

x2n?3dxdx?=?(x2?a2)n2(n?1)a2(x2?a2)n?12(n?1)a2?(x2?a2)n?1

21．

?dx1x?a=ln?C

x2?a22ax?a2（四）含有ax?b(a?0)的积分

?1arctan?dx?ab22．?2＝?ax?b?1ln?2?ab?23．

ax?Cb(b?0)

ax??b?C(b?0)ax??bx12dxlnax?b?C ＝?ax2?b2a2

x224．?xbdxax2?bdx＝a?a?ax2?b 25．?dx1x(ax2?b)＝

2blnx2ax2?b?C 26．

?dx1adxx2(ax2?b)＝?bx?b?ax2?b

27．?dxaax2?b1x3(ax2?b)＝2b2lnx2?2bx2?C 28．

?dxx1(ax2?b)2＝2b(ax2?b)??dx2bax2?b

（五）含有ax2?bx?c(a?0)的积分

?22ax??2arctanb29．?dx?4ac?b4ac?b2?Cax2?bx?c＝??12ax?b?b2?4??b2?4aclnac2ax?b?b2?4ac?C30．

?xax2?bx?cdx＝12bdx2alnax?bx?c?2a?ax2?bx?c（六）含有x2?a2(a?0)的积分 31．

?dx＝x2?a2arshxa?C1＝ln(x?x2?a2)?C 32．

?dx(x2?a2)3＝xa2x2?a2?C

33．

?xx2?a2dx＝x2?a2?C

34．

?x1(x2?a2)3dx＝?x2?a2?C

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(b2?4ac)(b2?4ac) 35．

??x2a22x?a?ln(x?x2?a2)?C dx＝22x2?a2x236．

x2(x2?a2)3dx＝?xx2?a2?ln(x?x2?a2)?C

1x2?a2?a?C 37．?＝ln22axxx?adx38．

?x?2x2?a2＝??C 222axx?adx2239．

x2a22x?a?ln(x?x2?a2)?C x?adx＝22x342222223(2x?5a)x?a?aln(x?x2?a2)?C (x?a)dx＝?88122(x2?a2)3?C 41．?xx?adx＝340．42．x?2xa42222x?adx＝(2x?a)x?a?ln(x?x2?a2)?C

882243．

?x2?a2?ax2?a222?C dx＝x?a?alnxx44．

?x2?a2x2?a2dx＝??ln(x?x2?a2)?C 2xx（七）含有x2?a2(a?0)的积分 45．

?dxx2?a2＝

xxarch?C1=lnx?x2?a2?C xa＝?46．

??dx(x?a)xx2?a2223xa2x?a22?C

47．

dx＝x2?a2?C

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48．

???x(x?a)x2223dx＝?1x?a22?C

49．

x2a22x?a?lnx?x2?a2?C dx＝22x2?a2x2(x2?a2)3dx＝?50．

xx2?a2?lnx?x2?a2?C

51．

?x?x?2dxx2?a2dx＝1aarccos?C ax52．

x2?a2＝?C 222axx?a2253．

x2a22x?a?lnx?x2?a2?C x?adx＝22x34222222223(2x?5a)x?a?alnx?x?a?C (x?a)dx＝?88122(x2?a2)3?C 55．?xx?adx＝354．56．x?2xa42222x?adx＝(2x?a)x?a?lnx?x2?a2?C

882257．

?ax2?a2dx＝x2?a2?aarccos?C

xxx2?a2x2?a222＝dx??lnx?x?a?C 2xx58．

?（八）含有a2?x2(a?0)的积分 59．

?dxa2?x2＝arcsinx?C a60．

?dx(a?x)223＝xa2a?x22?C

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