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高数积分公式大全

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常用积分公式

(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

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