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．?dxax?b＝

1alnax?b?C

2．?(ax?b)?dx＝3．?4．?xax?bdx＝

1a21a(??1)(ax?b)??1?C（???1）

(ax?b?blnax?b)?C

x2ax?bdxdx＝

1?1?22(ax?b)?2b(ax?b)?blnax?b?C 3?a?2??1blnax?bxab25．?x(ax?b)dx＝??C

6．?x(ax?b)x(ax?b)x222＝?1bx1a2?lnax?bx?C

7．?dx＝2(lnax?b?bax?b)?C

8．?(ax?b)dx＝

1a3(ax?b?2blnax?b?b2ax?b)?C

9．?dxx(ax?b)2＝

1b(ax?b)?1b2lnax?bx?C

（二）含有ax?b的积分 10．?ax?bdx＝

23a215a2(ax?b)?C

(3ax?2b)(ax?b)?C

(15ax?12abx?8b)(ax?b)?C

22233311．?xax?bdx＝12．?x13．?2ax?bdx＝

2105a3xax?bdx＝

23a2(ax?2b)ax?b?C

14．?x2ax?bdx＝

215a3(3ax?4abx?8b)ax?b?C

22215．??1ax?b?bln?C?ax?b?bdx?b＝?xax?b?2ax?barctan?C??b??b(b?0)

(b?0)16．?dxx2ax?b＝?ax?bbx?a2b?xdxdxax?b

17．?ax?bxax?bx2dx＝2ax?b?b?ax?bxa2

xax?bdxax?b18．?dx＝???x （三）含有x2?a2的积分 19．?20．?dxx?a22=

1aarctanxa?C

dx(x?a)dxx?a2222n=

x2(n?1)a(x?a)x?ax?a222n?1?2n?32(n?1)a2?(xdx2?a)2n?1

21．?=

12aln?C

（四）含有ax?b(a?0)的积分

??dx?＝?2ax?b??2?xax?b221ab1arctanabx?C?b?b(b?0)22．??ab1lnax?ax?

?C(b?0)23．?dx＝

2alnax?b?C

2224．?xdxax2?bdx＝

xba?a?ax2?b

225．?dxlnxx(ax2＝

1?b)2bax2?b?C

26．?dxadxx2(ax2＝?b)?1bx?b?ax2?b

27．dx2b?＝

ax3(ax2?b)2b2lnax?x2?12bx2?C

28．?dx＝

xdx(ax2?b)22b(ax2?b)?12b?ax2?b

（五）含有ax2?bx?c(a?0)的积分

?2ax?bCdx?4ac?b2arctan24ac?b2?29．?ax2?bx?c＝???1b2?4ac??C?b2ln2ax?b??4ac2ax?b?b2?4ac30．?xax2?bx?cdx＝

1lnax2?bx?c?b2a2a?dxax2?bx?c（六）含有x2?a2(a?0)的积分 31．?dx＝x2222arsha?C1＝ln(x?x?a)?C

x?a32．?dx＝x(x2?a2)3a2x2?a2?C

33．?xx＝x2?a2x2?a2d?C

34．?xdx＝?1(x2?a2)3x2?C

?a2(b2?4ac)(b2?4ac)

35．?x222dx＝x2x?a?22a2x?a2ln(x?22x?a)?C

36．?x2223dx＝?xx?a22?ln(x?22x?a)?C

(x?a)dxxx?adxx22237．?＝1alnx?a?axx?aax2222?C

2238．?x?a2222＝?2?C

39．?40．?x?adx＝

x2x?a?x8a222ln(x?x?a)?C

3822(x?a)dx＝

22223(2x?5a)x?a?(x?a)?C

223222aln(x?4x?a)?C

2241．?xx?adx＝42．?x22213x?adx＝

x82(2x?a)x?a?2222a48ln(x?x?a)?C

2243．?x?axx?ax2222dx＝x?a?aln2x?a?ax22?C

244．?dx＝?x?ax22?ln(x?x?a)?C

22（七）含有45．?dxx?a2x?a22(a?0)的积分

2＝xxarchxa?C1=lnx?xx?a22?C

46．?dx(x?a)223＝?a2x?a222?C

47．?xx?a22dx＝x?a?C

248．?x(x?a)x222223dx＝?1x?a22?C

49．?dx＝x2x?a?22a2x?a2lnx?x?a22?C

50．?x2223dx＝?xx?a22?lnx?x?a22?C

(x?a)dxxx?adxx22251．?＝

1aarccosax2?C

52．?x?a2222＝

x?aax22?C

53．?54．?x?adx＝x2x?a?x8222a222lnx?22x?a3822?C

22(x?a)dx＝

22223(2x?5a)x?a?(x?a)?C

223alnx?4x?a?C

55．?xx?adx＝56．?x22213x?adx＝

x8(2x?a)x?a?2222a48lnx?x?a22?C

57．?x?axx?ax2222dx＝x?a?aarccos22ax?C

258．?dx＝?x?ax22?lnx?x?a22?C

（八）含有a?x59．?dxa?xdx(a?x)22322(a?0)的积分

xa?C

22＝arcsin60．?＝ax2a?x22?C