高数积分公式大全

（一）含有axb的积分(a0)

dx1

1．＝lnaxbC

axba

2．(axb)dx＝

1

(axb)1C（1）

a(1)

x1

3．dx＝(axbblnaxb)C

axba2

x2114．dx＝(axb)22b(axb)b2lnaxbC

a32axbdx1axb

5．＝lnC

bxx(axb)

6．7．

dx1aaxb＝lnC

bxb2xx2(axb)

1bx

＝dx(lnaxb)C

(axb)2a2axb

x21b2

8．dx＝(axb2blnaxb)C

(axb)2a3axb

9．

dx11axb

＝lnCb(axb)b2xx(axb)2

（二）含有axb的积分

2

10．axbdx＝(axb)3C

3a2

11．xaxbdx＝(3ax2b)(axb)3C

15a2

2

12．x2axbdx＝(15a2x212abx8b2)(axb)3C

105a3

13．14．

2x

dx＝(ax2b)axbC

3a2axb

x22＝dx(3a2x24abx8b2)axbC15a3axb



dx

15．＝

xaxb



1lnbaxbb

C(b0)

axbb

2axbarctanC(b0)

bb

16．17．18．

x2

dxaxbadx

＝

bx2bxaxbaxb

axbdx

dx＝2axbbxxaxbaxbaxbadx

dx＝x2x2xaxb

（三）含有x2a2的积分19．

dx1x=arctanC

x2a2aa

dxx2n3dx

20．=

(x2a2)n2(n1)a2(x2a2)n12(n1)a2(x2a2)n1

21．

dx1xa

=lnCx2a22axa

（四）含有ax2b(a0)的积分

1

arctan

dxab

22．＝

ax2b1

ln

2ab

a

xCb

(b0)

axb

C(b0)

axb

x1

23．dx＝lnax2bC

ax2b2a

x2xbdx24．dx＝

ax2baaax2b

1x2dx

25．＝lnC

x(ax2b)2bax2b1adxdx

26．＝

x2(ax2b)bxbax2b

27．dxx＝alnax2bx1

C

3(ax2b)2b222bx2

28．

dxx1dx

(axb)＝

222b(ax2b)2bax2b

（五）含有ax2bxc(a0)的积分



22ax29．dx

arctanb4acb2

4acbC2

axc＝2bx

12axbb24ac

blnbbC

24ac2ax24ac30．xaxdx＝1lnax2bxc

b2bxc2a2adx

ax2bxc

（六）含有x2a2(a0)的积分31．dx

＝arshx

x2a2aC＝ln(xx21a2)C

32．dx＝

x(xC

2a2)3

a2x2a2

33．xxdx＝x2a2C

2a2

34．

x(xdx＝

12a2)3

xC

2a2

35．

x2

xxa2a2

2ln(xxdx＝2a2

22x2a2)C36．

x2(x2adx＝

xln(xx2a2)C

2)3

x2a2

37．dx

＝1x2a2a

xx2a2

alnxC

38．

dx

＝x2a2x2

x2a2

aC

2x39．

x2a2dx＝xa2

2x2a22

ln(xx2a2)C(b24ac)(b24ac)

x3

40．(x2a2)3dx＝(2x25a2)x2a2a4ln(xx2a2)C

881

41．xx2a2dx＝(x2a2)3C

3

42．x243．

xa4

x2a2dx＝(2x2a2)x2a2ln(xx2a2)C

88

x2a2x2a2a

dx＝x2a2alnCxxx2a2x2a2

dx＝ln(xx2a2)Cx2x

44．

（七）含有x2a2(a0)的积分45．46．47．48．

xx

＝archC=lnxx2a2C

1xax2a2

dx

dx(x2a2)3

xx2a2

＝

xa2x2a2

C

dx＝x2a2C

x(x2a2)3

dx＝

1x2a2

C

49．

xa2

x2a2lnxx2a2Cdx＝22x2a2

x2

50．

x2(x2a2)3

dx

dx＝

xx2a2

lnxx2a2C

1a

51．＝arccosC

axxx2a2

52．

x2

x2a2＝C

a2xx2a2

dx

53．．

xa2

x2a2lnxx2a2Cx2a2dx＝22

x3

(x2a2)3dx＝(2x25a2)x2a2a4lnxx2a2C

88

1

55．xx2a2dx＝(x2a2)3C

3

56．x257．

xa4

x2a2dx＝(2x2a2)x2a2lnxx2a2C

88

ax2a2

22＝xaaarccosCdx

xx

x2a2x2a2

dx＝lnxx2a2Cx2x

58．

（八）含有a2x2(a0)的积分59．60．61．62．

x

＝arcsinC

aa2x2dx

dx(a2x2)3

xa2x2

＝

xa2a2x2

C

dx＝a2x2C

x(a2x2)3

dx＝

1a2x2

C

63．

xa2x22axarcsinCdx＝22aa2x2

x

arcsinCdx＝

aa2x2(a2x2)3

dx

x2

．

x2x

1aa2x2

65．＝lnC

axxa2x2

66．

x2

a2x2＝C

a2xa2x2

dx

67．

xa2xa2x2arcsinCa2x2dx＝22a

x3x

68．(a2x2)3dx＝(5a22x2)a2x2a4arcsinC

88a1

69．xa2x2dx＝(a2x2)3C

3

70．x271．72．

xa4x

a2x2dx＝(2x2a2)a2x2arcsinC

88a

a2x2aa2x2

dx＝a2x2alnCxxa2x2a2x2x

dx＝arcsinCx2xa

（九）含有ax2bxc(a0)的积分73．74．75．76．77．78．

1

＝ln2axb2aax2bxcC

aax2bxc

dx

2axb

ax2bxcdx＝ax2bxc

4a

1dx＝ax2bxc

aax2bxc

xdxcbxax2

＝

12axbarcsinCab24ac

2axbb24ac2axb

22＝cbxaxdxcbxaxarcsinC

4a8a3b24ac

1b2axb

dx＝cbxax2arcsinC

acbxax22a3b24ac

x

xa

（十）含有或(xa)(bx)的积分

xb

79．80．81．

xaxa

dx＝(xb)(ba)ln(xbxb

xaxb)C

xaxa

＝dx(xb)(ba)arcsin

bxbxxa

Cbx

dx

＝2arcsin

(xa)(bx)

xa

Cbx

(ab)

82．

2xab(ba)2xa

(xa)(bx)arcsinC(xa)(bx)dx＝

44bx

（十一）含有三角函数的积分

83．sinxdx＝cosxC84．cosxdx＝sinxC85．tanxdx＝lncosxC86．cotxdx＝lnsinxC

87．secxdx＝lntan()C＝lnsecxtanxC

x

88．cscxdx＝lntanC＝lncscxcotxC

2

4

x2

．sec2xdx＝tanxC90．csc2xdx＝cotxC91．secxtanxdx＝secxC92．cscxcotxdx＝cscxC

x1

93．sin2xdx＝sin2xC

24x1

94．cos2xdx＝sin2xC

241n1

sinn2xdx95．sinnxdx＝sinn1xcosx

nn1n1

cosn2xdx96．cosnxdx＝cosn1xsinx

nn

dx1cosxn2dx

97．＝

sinnxn1sinn1xn1sinn2xdx1sinxn2dx

98．＝

cosnxn1cosn1xn1cosn2x

1m1

m1n1cosm2xsinnxdx99．cosmxsinnxdx＝cosxsinx

mnmn

1n1

cosmxsinn2xdx＝cosm1xsinn1x

mnmn

100．sinaxcosbxdx＝

11

cos(ab)xcos(ab)xC

2(ab)2(ab)

11

101．sinaxsinbxdx＝sin(ab)xsin(ab)xC

2(ab)2(ab)

11

102．cosaxcosbxdx＝sin(ab)xsin(ab)xC

2(ab)2(ab)

x

atanb2dx2103．＝arctanC

absinxa2b2a2b2

(a2b2)

x

2a2atanbbdx12104．＝lnC

absinxb2a2atanxbb2a2

2

2ababxdx

105．＝arctan(tan)C

ababab2abcosx

(a2b2)

(a2b2)

x

tan

dx1ab2106．＝lnabcosxabbax

tan2abbaCabba

(a2b2)

dx1b

107．＝arctan(tanx)C

a2cos2xb2sin2xaba

108．

dx1btanxa

＝lnC2abbtanxaa2cos2xb2sin2x

11

109．xsinaxdx＝sinaxxcosaxC

a2a122

110．x2sinaxdx＝x2cosaxxsinaxcosaxC

aa2a311

111．xcosaxdx＝cosaxxsinaxC

a2a122

112．x2cosaxdx＝x2sinaxxcosaxsinaxC

aa2a3

（十二）含有反三角函数的积分（其中a0）113．arcsindx＝xarcsina2x2C

x2a2xxx

114．xarcsindx＝()arcsina2x2C

24a4ax3x1x

115．x2arcsindx＝arcsin(x22a2)a2x2C

3a9a

x

axa

xx

116．arccosdx＝xarccosa2x2C

aa

x2a2xxx

117．xarccosdx＝()arccosa2x2C

24a4ax3x1x

118．x2arccosdx＝arccos(x22a2)a2x2C

3a9a

xxa

119．arctandx＝xarctanln(a2x2)C

aa2x1xa

120．xarctandx＝(a2x2)arctanxC

a2a2

x3xaa3x

121．x2arctandx＝arctanx2ln(a2x2)C

3a66a

（十三）含有指数函数的积分

1

122．axdx＝axC

lna1

123．eaxdx＝eaxC

a1

124．xeaxdx＝(ax1)eaxC

a21n

125．xneaxdx＝xneaxxn1eaxdx

aa

x1

126．xaxdx＝axaxC

lna(lna)2

1n

nxxn1axdxxalnalna

1

128．eaxsinbxdx＝eax(asinbxbcosbx)C

a2b2

1

129．eaxcosbxdx＝eax(bsinbxacosbx)C

a2b2

1

130．eaxsinnbxdx＝eaxsinn1bx(asinbxnbcosbx)

a2b2n2

1

131．eaxcosnbxdx＝eaxcosn1bx(acosbxnbsinbx)

a2b2n2

127．xnaxdx＝

（十四）含有对数函数的积分132．lnxdx＝xlnxxC

dx

133．＝lnlnxC

xlnx

11

134．xnlnxdx＝xn1(lnx)C

n1n1

135．(lnx)ndx＝x(lnx)nn(lnx)n1dx

136．xm(lnx)ndx＝1n

m1xm1(lnx)n

m1

xm(lnx)n1dx（十五）含有双曲函数的积分137．shxdx＝chxC138．chxdx＝shxC139．thxdx＝lnchxC140．sh2xdx＝x12

4

sh2xC

141．ch2xdx＝x21

4

sh2xC

（十六）定积分

142．cosnxdx＝sinnxdx＝0





143．cosmxsinnxdx＝0



144．

cosmxcosnxdx＝

0,mn

,mn

145．

sinmxsinnxdx＝

0,mn

,mn

0,mn

146．sinmxsinnxdx＝cosmxcosnxdx＝

002

,mn147．I＝

sin2nxdx＝2n

cosnxdx

n01

0

I＝

nnI

n2

In1n3n42

nn2L5

3

（n为大于1的正奇数），Inn1nn3n2L3142

2（n为正偶数），I＝

02

I＝11