高等数学常用导数积分公式查询表好

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08070141常用导数和积分公式

导数公式：

? (1) (C)?0 ? (3) (sinx)?cosx

???1?(x)??x (2)

? (4) (cosx)??sinx

(5)

(tanx)??sec2x (7) (secx)??secxtanx

(9)

(ax)??axlna (log1 (11)

ax)??xlna

(arcsinx)??1 (13)

1?x2

(arctanx)??1 (15)

1?x2

(cotx)???csc2x (cscx)???cscxcotx

(ex)??ex

(lnx)??1x，

(arccosx)???11?x2(arccotx)???11?x2

(6)

(8) (10) (12)

(14)

(16)

08070141常用导数和积分公式

基本积分表

?tgxdx??lncosx?C?ctgxdx?lnsinx?C?secxdx?lnsecx?tgx?C?cscxdx?lncscx?ctgx?Cdx1x?arctg?C?a2?x2aadx1x?a?ln?x2?a22ax?a?Cdx1a?x??a2?x22alna?x?Cdxx?arcsin?C?a2?x2a?2ndx2?cos2x??secxdx?tgx?Cdx2?sin2x??cscxdx??ctgx?C?secx?tgxdx?secx?C?cscx?ctgxdx??cscx?Cax?adx?lna?Cx?shxdx?chx?C?chxdx?shx?C?dxx2?a2?ln(x?x2?a2)?C?2In??sinxdx??cosnxdx?00n?1In?2n???

x2a22x?adx?x?a?ln(x?x2?a2)?C22x2a2222x?adx?x?a?lnx?x2?a2?C22x2a2x222a?xdx?a?x?arcsin?C22a22三角函数的有理式积分：

2u1?u2x2dusinx?，　cosx?，　u?tg，　dx? 22221?u1?u1?u（一）含有ax?b的积分(a1．

?0)

dx1＝?ax?balnax?b?C

??(ax?b)dx＝

2．

1(ax?b)??1?C（???1）

a(??1)08070141常用导数和积分公式

3．

x1dx＝?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．

dx?x(ax?b)＝?1ax?bln?C bx6．

?dx1aax?b＝??ln?C 22x(ax?b)bxbx7．

1bx(lnax?b?)?C dx＝2?(ax?b)2aax?b1b2x2)?C 8．?dx＝3(ax?b?2blnax?b?2aax?b(ax?b)9．

?dx11ax?b＝?ln?C

x(ax?b)2b(ax?b)b2x（二）含有10．

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 ax?bdx＝12．?x3105aax?bdx＝13．

?2xdx＝2(ax?2b)ax?b?C

3aax?b2x2(3a2x2?4abx?8b2)ax?b?C dx＝315aax?b14．

???dx?15．?＝?xax?b???1lnbax?b?b?C(b?0)ax?b?b

2ax?barctan?C(b?0)?b?b08070141常用导数和积分公式

16．

?x2dxax?badx＝???bx2bxax?bax?b

17．

??dxax?b dx＝2ax?b?b?xxax?b18．

ax?bax?badx＝dx??x2x2?xax?b2 （三）含有x19．

?a2的积分

dx?x2?a21xarctan?C aa

20．

dxx2n?3dx?=?(x2?a2)n2(n?1)a2(x2?a2)n?12(n?1)a2?(x2?a2)n?121．

?dx1x?a=ln?C

x2?a22ax?a2（四）含有ax?b(a?0)的积分

ax?Cb(b?0)

?1arctan?dx?ab22．?＝?ax2?b?1ln?2?ab?23．

ax??b?C(b?0)ax??bx12dx＝?ax2?b2alnax?b?C

xbdxx2?dx24．?＝ aa?ax2?bax2?b25．

?1x2dxln?C ＝

x(ax2?b)2bax2?b26．

1adxdx??＝?x2(ax2?b)bxb?ax2?b

08070141常用导数和积分公式

27．?dxaax2?b1x3(ax2?b)＝2b2lnx2?2bx2?C 28．

?dxdx(ax2?b)2＝

x12b(ax2?b)?2b?ax2?b （五）含有ax2?bx?c(a?0)的积分

?22ax?b．?dx??4ac?b2arctan4ac?b2?C29ax2?bx?c＝??12ax?b?2??b2?4aclnb?4ac2ax?b?b2?4ac?C30．

?xax2?bx?cdx＝12alnax2?bx?c?bdx2a?ax2?bx?c（六）含有x2?a2(a?0)的积分

31．

?dx＝arshxx2?a2a?C1＝ln(x?x2?a2)?C 32．

?dxx(x2?a2)3＝

a2x2?a2?C

33．

?xx2?a2dx＝x2?a2?C

34．

?x?1(x2?a2)3dx＝x2?a2?C

35．

?x2x2x2?a2dx＝2x2?a2?a2ln(x?x2?a2)?C 36．

?x2(x2?a2)3dx＝?xx2?a2?ln(x?x2?a2)?C

37．

?dx1x2?a2?axx2?a2＝alnx?C (b2?4ac)(b2?4ac)

08070141常用导数和积分公式

103．

dx?a?bsinxdx?a?bsinx＝2a2?b2atanarctanx?b2?C22a?b(a2?b2)

104．＝x22?b?b?a12ln?C22x22b?aatan?b?b?a2atan(a2?b2)

105．

dx2a?ba?bx＝arctan(tan)?C?a?bcosxa?ba?ba?b2(a2?b2)

x?dx1a?b2106．?＝lna?bcosxa?bb?axtan?2tan107．

a?bb?a?Ca?bb?a(a2?b2)

dx1barctan(tanx)?C ＝?a2cos2x?b2sin2xabadx1btanx?a＝ln?a2cos2x?b2sin2x2abbtanx?a?C

108．

11sinax?xcosax?C 2aa12222110．?xsinaxdx＝?xcosax?2xsinax?3cosax?C

aaa11111．?xcosaxdx＝2cosax?xsinax?C

aa12222112．?xcosaxdx＝xsinax?2xcosax?3sinax?C

aaa（十二）含有反三角函数的积分（其中a?0）

xx22113．?arcsindx＝xarcsin?a?x?C

aa109．

?xsinaxdx＝

xx2a2xx2?)arcsin?a?x2?C 114．?xarcsindx＝(a24a4xx3x1arcsin?(x2?2a2)a2?x2?C 115．?xarcsindx＝

a3a9208070141常用导数和积分公式

116．

xxarccosdxxarccos?a2?x2?C ＝?aaxx2a2xx2?)arccos?a?x2?C 117．?xarccosdx＝(a24a4xx3x1arccos?(x2?2a2)a2?x2?C 118．?xarccosdx＝

a3a92xxa22arctandxxarctan?ln(a?x)?C ＝?aa2x12xa2120．?xarctandx＝(a?x)arctan?x?C

a2a2119．

xx3xa2a3arctan?x?ln(a2?x2)?C 121．?xarctandx＝

a3a662（十三）含有指数函数的积分

1xa?C lna1axax123．?edx＝e?C

a1axax124．?xedx＝2(ax?1)e?C

a1naxnn?1axnax125．?xedx＝xe??xedx

aa122．

xa?dx＝

126．

x?xadx＝

xx1a?ax?C 2lna(lna)1nxnxa?xn?1axdx ?lnalna1axaxe(asinbx?bcosbx)?C 128．?esinbxdx＝22a?b1axeax(bsinbx?acosbx)?C 129．?ecosbxdx＝22a?b1axneaxsinn?1bx(asinbx?nbcosbx) 130．?esinbxdx＝222a?bn127．

nx?xadx＝

n(n?1)b2axn?2?2esinbxdx a?b2n2?08070141常用导数和积分公式

131．

axn?ecosbxdx＝

1eaxcosn?1bx(acosbx?nbsinbx) 222a?bn

n(n?1)b2ax?2ecosn?2bxdx 22?a?bn（十四）含有对数函数的积分 132．133．

?lnxdx＝xlnx?x?C

dx?xlnx＝lnlnx?C

1n?11nx(lnx?)?C 134．?xlnxdx＝

n?1n?1135．136．

nn(lnx)dx＝x(lnx)?n?(lnx)dx ?mn?x(lnx)dx＝

n?11nmn?1xm?1(lnx)n?x(lnx)dx ?m?1m?1（十五）含有双曲函数的积分 137．138．139．140．