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 ＝22bxbxx(ax?b)7．
x1bdx＝(lnax?b?)?C ?(ax?b)2a2ax?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 ?3a2311．?xax?bdx＝(3ax?2b)(ax?b)?C 215a22(15a2x2?12abx?8b2)(ax?b)3?C 12．?xax?bdx＝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??2dxax?badx＝? ??bx2bxax?bax?b17．
ax?bdxdx＝2ax?b?b? xxax?bax?bax?badxdx＝ ??x2x2?xax?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
?1arctan?dx?ab22．?2＝?ax?b?1ln?2?ab?23．
ax?Cb(b?0)
ax??b?C(b?0)ax??bx12dx＝?ax2?b2alnax?b?C
24．?x2ax2?bdx＝xa?ba?dxax2?b
．?dxx(ax2?b)＝1x2252blnax2?b?C
26．
?dx1ax2(ax2?b)＝?bx?dxb?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?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．
?dxx2?a2＝arshxa?C1＝ln(x?x2?a2)?C 32．
?dxx(x2?a2)3＝a2x2?a2?C
33．
?xx2?a2dx＝x2?a2?C
34．
?x1(x2?a2)3dx＝?x2?a2?C
(b2?4ac)(b2?4ac)
高等数学积分公式和微积分公式大全
