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．?a?2ax?b?5．
1ax?bdx?＝?x(ax?b)blnx?C
6．
?1aax?bdx??ln?C ＝
bxb2xx2(ax?b)7．
x1bdx＝(lnax?b?)?C 2?(ax?b)2aax?bx21b2dx＝3(ax?b?2blnax?b?)?C 8．?2(ax?b)aax?b9．
?11ax?bdx?ln?C ＝
xx(ax?b)2b(ax?b)b2（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．
??x2dx＝2(ax?2b)ax?b?C
3aax?bx22222dx＝(3ax?4abx?8b)ax?b?C 315aax?b14．
??dx?15．?＝?xax?b???16．
1lnbax?b?b?C(b?0)ax?b?b
2ax?barctan?C(b?0)?b?b?x??2dxax?badx＝? ??bx2bxax?bax?b17．
ax?bdxdx＝2ax?b?b? xxax?bax?bax?badxdx＝ ??2?xx2xax?b2218．
（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
?1arctana22．?dx??abbx?C(b?0)ax2?b＝??1ax??b
?lnax??b?C(b?0)?2?ab23．
?xax2?bdx＝122alnax?b?C
24．?x2ax2?bdx＝xbdxa?a?ax2?b
25．?dx1x2x(ax2?b)＝2blnax2?b?C 26．
?dx1x2(ax2?b)＝?bx?adxb?ax2?b
27．?dxaax2?b1x3(ax2?b)＝2b2lnx2?2bx2?C
28．
?dxx(ax2?b)2＝2b(ax2?b)?1dx2b?ax2?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．
?x12bdxax2?bx?cdx＝2alnax?bx?c?2a?ax2?bx?c（六）含有x2?a2(a?0)的积分 31．?dxxx2?a2＝arsh?C1＝ln(x?x2?a2a)?C
(b2?4ac)(b2?4ac)
32．
??dx(x?a)xx2?a2223＝xa2x?a22?C
33．
dx＝x2?a2?C
1x?a2234．
?x(x?a)223dx＝??C
35．
??x2a22x?a?ln(x?x2?a2)?C dx＝22x2?a2x236．
x2(x2?a2)3dxdx＝?xx2?a2?ln(x?x2?a2)?C
1x2?a2?a?C 37．?＝ln22axxx?a38．
?x2x2?a2?C ＝?222axx?adx39．
?x2a22x?a?ln(x?x2?a2)?C x?adx＝2222x34223222222(x?a)dx＝(2x?5a)x?a?aln(x?x?a)?C ?8812241．?xx?adx＝(x2?a2)3?C
340．42．x?2xa42222x?adx＝(2x?a)x?a?ln(x?x2?a2)?C
882243．
??x2?a2x2?a2?a22dx＝x?a?aln?C xxx2?a2x2?a222dx??ln(x?x?a)?C ＝2xx44．
（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
1x?a2248．
?x(x?a)223dx＝??C
49．
???xx2a22x?a?lnx?x2?a2?C dx＝22x2?a2x250．
x2(x2?a2)3dxx2?a2dxdx＝?xx2?a2?lnx?x2?a2?C
51．＝1aarccos?C ax52．
?x2x2?a2?C ＝222axx?a2253．
?x2a22x?a?lnx?x2?a2?C x?adx＝22x34223222222(x?a)dx＝(2x?5a)x?a?alnx?x?a?C ?8812255．?xx?adx＝(x2?a2)3?C
354．56．x?2xa42222x?adx＝(2x?a)x?a?lnx?x2?a2?C
8822
高数积分公式大全
