常 用 积 分 公 式
（一）含有ax?b的积分(a?0) 1．?dx1＝lnax?b?C ax?ba1(ax?b)??1?C（???1）
a(??1)2．?(ax?b)?dx＝3．?x1dx＝2(ax?b?blnax?b)?C ax?ba1?1x2?dx＝3?(ax?b)2?2b(ax?b)?b2lnax?b??C 4．?a?2ax?b?5．?6．?7．?1ax?bdx?C ＝?lnbxx(ax?b)1aax?bdx??ln?C ＝
bxb2xx2(ax?b)x1bdx＝(lnax?b?)?C 22(ax?b)aax?bx21b2dx＝3(ax?b?2blnax?b?)?C 8．?(ax?b)2aax?b9．?11ax?bdx?ln?C ＝22b(ax?b)bxx(ax?b)（二）含有ax?b的积分
2(ax?b)3?C 3a2(3ax?2b)(ax?b)3?C 11．?xax?bdx＝215a22223(15ax?12abx?8b)(ax?b)?C 12．?x2ax?bdx＝3105a10．?ax?bdx＝13．?14．?x2dx＝2(ax?2b)ax?b?C
3aax?bx22dx＝(3a2x2?4abx?8b2)ax?b?C 315aax?b??dx?15．?＝?xax?b???1lnbax?b?b?C(b?0)ax?b?b
2ax?barctan?C(b?0)?b?b16．?17．?18．?x2dxax?badx＝? ??bx2bxax?bax?bax?bdxdx＝2ax?b?b? xxax?bax?bax?badxdx＝ ??2?xx2xax?b（三）含有x2?a2的积分 19．?20．?21．?dx1x=arctan?C x2?a2aadxx2n?3dx?=
(x2?a2)n2(n?1)a2(x2?a2)n?12(n?1)a2?(x2?a2)n?11x?adxln?C =22x?a2ax?a（四）含有ax2?b(a?0)的积分
?1arctan?dx?ab22．?2＝?ax?b?1ln?2?ab?ax?Cb(b?0)ax??b?C(b?0)ax??b
23．?x12＝dxlnax?b?C 2ax?b2ax2xbdx24．?2dx＝??2
ax?baaax?bdx1x2?C 25．?＝ln2x(ax2?b)2bax?b26．?dx1adx??＝
x2(ax2?b)bxb?ax2?b27．?dxaax2?b1x3(ax2?b)＝2b2lnx2?2bx2?C
28．?dxx1dx(ax2?b)2＝2b(ax2?b)?2b?ax2?b （五）含有ax2?bx?c(a?0)的积分
?22ax?29．?dx??4ac?b2arctanb4ac?b2?Cax2?bx?c＝??12ax?b?b2?4ac??b2?4acln2ax?b?b2?4ac?C30．?x1bdxax2?bx?cdx＝2alnax2?bx?c?2a?ax2?bx?c（六）含有x2?a2(a?0)的积分 31．?dxxx2?a2＝arsha?C1＝ln(x?x2?a2)?C 32．?dx(x2?a2)3＝xa2x2?a2?C
33．?xx2?a2dx＝x2?a2?C
34．?x(x2?a2)3dx＝?1x2?a2?C
35．?x2xx2?a2dx＝2x2?a?a222ln(x?x2?a2)?C 36．?x2(x2?a2)3dx＝?xx2?a2?ln(x?x2?a2)?C
．?dx1x2?a237?axx2?a2＝alnx?C 38．?dxx2?a2x2x2?a2＝?a2x?C 39．?x2?a2dx＝x2x2?a2?a22ln(x?x2?a2)?C (b2?4ac)(b2?4ac)
40．?(x2?a2)3dx＝(2x2?5a2)x2?a2?a4ln(x?x2?a2)?C 41．?xx2?a2dx＝42．?x43．?44．?222x8381(x2?a2)3?C 3xa42222x?adx＝(2x?a)x?a?ln(x?x2?a2)?C
88x2?a2x2?a2?a22dx＝x?a?aln?C xxx2?a2x2?a222dx??ln(x?x?a)?C ＝2xx（七）含有x2?a2(a?0)的积分 45．?46．?47．?48．?49．?50．?51．?52．?53．?dxx2?a2＝
xxarch?C1=lnx?x2?a2?C xadx(x?a)xx2?a2223＝?xa2x?a22?C
dx＝x2?a2?C
x(x?a)223dx＝?1x?a22?C
x2a22x?a?lnx?x2?a2?C dx＝22x2?a2x2x2(x2?a2)3dxxx2?a2dx＝?1axx2?a2?lnx?x2?a2?C
＝arccosa?C xx22x2?a2?C ＝222axx?adxx2a22x?a?lnx?x2?a2?C x?adx＝22254．?(x2?a2)3dx＝(2x2?5a2)x2?a2?a4lnx?x2?a2?C
x83855．?xx2?a2dx＝56．?x57．?58．?2221(x2?a2)3?C 3xa42222x?adx＝(2x?a)x?a?lnx?x2?a2?C
88ax2?a2dx＝x2?a2?aarccos?C
xxx2?a2x2?a222dx??lnx?x?a?C ＝2xx（八）含有a2?x2(a?0)的积分 59．?60．?61．?62．?63．?64．?dxa2?x2＝arcsin?C ＝xa2xadx(a?x)xa2?x2223a?x22?C
dx＝?a2?x2?C
x(a?x)223dx＝1a?x22?C
x2a2x2a?x?arcsin?C dx＝?22aa2?x2x2x2(a2?x2)3dxdx＝xa2?x2?arcsinx?C a1a?a2?x2?C 65．?＝ln22axxa?x66．?67．?x22a2?x2?C ＝?222axa?xdxx2a2x2a?x?arcsin?C a?xdx＝22a2x8169．?xa2?x2dx＝?(a2?x2)3?C
368．?(a2?x2)3dx＝(5a2?2x2)a2?x2?a4arcsin?C
38xa
高数积分公式大全
