高等数学积分公式大全

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．

x1＝dx?ax?ba2(ax?b?blnax?b)?C

x21?1?dx＝3?(ax?b)2?2b(ax?b)?b2lnax?b??C 4．?ax?ba?2?5．

?dx1ax?b＝?ln?C

bxx(ax?b)6．

dx1aax?b＝??ln?C ?x2(ax?b)bxb2xx1b＝dx(lnax?b?)?C ?(ax?b)2a2ax?b7．

x21b2dx＝3(ax?b?2blnax?b?)?C 8．?(ax?b)2aax?b9．

dx11ax?b?ln?C ＝?x(ax?b)2b(ax?b)b2x（二）含有ax?b的积分

23(ax?b)?C ?3a2311．?xax?bdx＝(3ax?2b)(ax?b)?C 215a2212．?xax?bdx＝(15a2x2?12abx?8b2)(ax?b)3?C 3105a10．

ax?bdx＝13．

?x2dx＝2(ax?2b)ax?b?C

3aax?b14．

?x22dx＝(3a2x2?4abx?8b2)ax?b?C 315aax?b??dx?15．?＝?xax?b???16．

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

2ax?barctan?C(b?0)?b?b?x??2ax?badxdx?＝? bx2b?xax?bax?b17．

ax?bdxdx＝2ax?b?b? xxax?bax?badxax?b??dx＝

x2?xax?bx22218．

（三）含有x?a的积分 19．

dx1x=arctan?C ?x2?a2aadxx2n?3dx=??(x2?a2)n2(n?1)a2(x2?a2)n?12(n?1)a2?(x2?a2)n?1 1x?adxln=?x2?a22ax?a?C

220．

21．

（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??bx12＝dx?ax2?b2alnax?b?C

24．?x2xbdxax2?bdx＝a?a?ax2?b 25．?dx1x(ax2?b)＝

2blnx2ax2?b?C 26．

?dx1ax2(ax2?b)＝?bx?dxb?ax2?b

27．?dxax3(ax2?b)＝2b2lnax2?bx2?12bx2?C 28．

?dx(ax2?b)2＝x2b(ax2?b)?1dx2b?ax2?b

（五）含有ax2?bx?c(a?0)的积分

?22ax?2arctan?b29．?dx?4ac?b4ac?b2?Cax2?bx?c＝??12ax?b?b2??ln4ac?C?b2?4ac2ax?b?b2?4ac30．

?x1bdxax2?bx?cdx＝2alnax2?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．

?dxx(x2＝?a2)3a2x2?a2?C

33．

?x2x2?a2dx＝x?a2?C

34．

?x1(x2?a2)3dx＝?x2?a2?C

(b2?4ac)(b2?4ac) 35．

??x2a22dx＝x?a?ln(x?x2?a2)?C

22x2?a2x2(x2?a2)3dx＝?x236．

xx2?a2?ln(x?x2?a2)?C

1x2?a2?a?C 37．?＝ln22axxx?adx38．

?x2x2?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?88141．?xx2?a2dx＝(x2?a2)3?C

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 2xx22（七）含有x?a(a?0)的积分 45．

??dxx2?a2dx＝xxarch?C1=lnx?x2?a2?C xa＝?46．

xa2(x?a)223x?a22?C

47．

?xx2?a2dx＝x2?a2?C

48．

?x(x?a)223dx＝?1x?a22?C

49．

??x2a22dx＝x?a?lnx?x2?a2?C

22x2?a2x2(x2?a2)3dx＝?x250．

xx2?a2?lnx?x2?a2?C

51．

?x?x2dxx2?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?88155．?xx2?a2dx＝(x2?a2)3?C

354．56．x??2xa42222x?adx＝(2x?a)x?a?lnx?x2?a2?C

882257．

ax2?a2dx＝x2?a2?aarccos?C

xxx2?a2x2?a222dx??lnx?x?a?C ＝2xx2258．

?（八）含有a?x(a?0)的积分 59．

??dxa2?x2dx＝arcsinx?C ax?C

60．

(a?x)223＝a2a?x22

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