考研常用积分公式（范文篇）

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考研常用积分公式(D

(x刈

="ji-1

(ax)/

=axIna

(e

x)

/

=ex

(log/

lax)=xIna

(Inx)/

l=x

(sinx)/

=cosx

(cosX)/

="sinx

(tanx)/

=sec2x

(cotx)/

=-csc2x

(secx)/

=secxtanx

(escx)/

=-cscxcotx

(aresinx

)

/

(arccosx

)/

=(larctanx)/

=l+x2

(arccotx)/=-ll+x2

(uv)

/

=yU/V+UV/

lu)/.

u/v-uv/\vIJ=v2Jkdx=kx

xN+1

dx=|i+lJdx

x=lnxJdx

1+x2=arctanx

=arcsinx

Jcosxdx=sinx

Jsinxdx=-cosxJsec2xdx=tanxJcsc2

xdx="cotx

Jsecxtanxdx=secxJescxcotxdx=-cscxJex

dx=e

Jax

dx=ax

Ina

Jtanxdx="lncosxJcotxdx=lnsinx

fsecxdx=lnsecx+tanx

fescxdx=lnescx-cotx

fllx2+a2

dx=aarctanx

aJllx-x2-a2dx=2

aInax+a

=lnx

x=arcsina

等价无穷小(x—0)sinx-x

tanx-x

arcsinx-x

arctanx-xln(l+x)-xex-l-x

121-cos

x-2x

ll~2x

ax-1〜xIna

f(x)渐近线k=limx—coxb=

limx一8

FLf(x)-kx1J

y〃

曲率k=

(

31+y

/2)

2

1

考研常用积分公式(2)

(x)

(arctanx)=(arccotx)

11+x

2

Jcscxdx="cotx

2

2

l=arcsin

(a)

=aIna=e

/x

11+x

\secxtanxdx=Jescxcotxdx=Jedx=eJadx

secx

等价无穷小(x-0)

-CSCX

sinx~x

tanx~x

(e)

x

/

(uv)

lxIna

/

=UV+uv

(logax)(lnx)

arcsinx〜x

(sinx)

Jkdx=kx

Ina

arctanx-x

=cosx

(cosx)(tanx)(cotx)(secx)(cscx)

/

=-sinx=secx="cscx=secxtanx="cscxcotx

22

xdx=

N

x

N+I

N+I

Jtanxdx="lncosxJcotxdx=Jsecxdx=

InsinxInsecx+tanx

ln(l+x)-xe-1〜x

dx

/

J1+x

xdx

=lnx=arctanx

1-cosx〜

12

12x

2

2

/

J

Jescxdx=lnescx-cotx

=arcsinx

1〜

x

/

Jxjx

1

2

+a1

2

dx=

la

arctan

xa

a-l-xIna

(aresinx

)

Jcosxdx=

sinx

2

-a

2

dx=

12a

In

x-ax+

a

渐近线k=lim

f

(x)

X

X—>00

Jsinxdx=

2

-COSX

(arccosx

)

/

1\secxdx=tanx

f

=lnx+

b=limFLf(x)-kxlJx一oo

最新下载(NewD)中国最大、最专业的学习资

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曲率k=

//

32

(1+y)

/2

最新下载(NewD)中国最大、最专业的学习资

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考研常用积分公式(3)

当x—>0时x〜sinx〜tanx-arcsinx-arctanx-ln(l+x)-

2.常用极限

1.lim

nk

x—>ooax—>oo

ax-Una

1+xb-1

b

(其中a>0,b^O)

x~ex-1x2~l-cosxx—1a〜(1+x)a

2

n

11

n

=0,(a>D

2.lim

cn

x一oon!x—>oo

=0,(c>0)

3.limnqn,(qx一oo

n

4.limn=1,(a>0)6.lim

logann

x—>oo

=0,(a>l)

In

7.lim9.lim

x0e

8.liml+=e

x一oo

n

x10.lim

sinxx

x一oo

=0

)=

lp+1

11.lim

logaxxE

X一+oo

=0,(a>l,E>0)

np+1

12.lim(

x—>oo

Ip+2p+-+np

np+1

2pp+1

13.lim(

x一oo

Ip+2p+…+np

np+lln+l

ln+2

=212n

1

14.lim(

x一oo

Ip+3p+,+(2n-l)p

np+1

=15.lim

X—00

++•••+

=ln2

p

17.lim(1+x)=e

x—*0

1

x

16.lim

sinxxax-lx

x—>0

=1=

am

18.lim20.lim22.lim24.lim26.lim28.x—0

=lna=1=1

19.lim21.lim23.lim25.lim

(l+a)|i-l

aarcsinxx

x—>0

ln(l+x)

xarctanx

x

m

x一Ox—>0

=1

=mn(n-m)

21

(1+mx)n-(1+nx)m

x2

m

x一Ox—>0

n

-xx—>0

一,(mn#)

n

n

•la

xmx—>0

+,(mn)

P

xm-lxn-1

mx一1

mnn

,(m,n为自然数)，(m,n£Z)

27.lim

x一1

m1-xm

n1-xn

m-n2

x—>m

29.若Xn(n=l,2...)收敛，则算数平均值的序列，n=

X1+X2+Xn,(n=l,2)也收敛，且

n

1

lim

xl+x2+-+xn

n

x—>00

=limxn

X—00

30.若序列Xn(n=l,2...)收敛，且Xn>0,贝！1lim=limXn

x一oo

X一00

n

31.若Xn>0(n=l,2...)且lim

Xn+1

x一ooXn

lim=lim

x—>oo

n

Xn+1

x—>ooXn

32.若整序变量Yn->+oo,并且—至少是从某一项开始

----在n增大时Yn亦增大，Yn+l>Yn,则

n一ooYn

lim

Xn

=lim

Xn-Xn-1

n-ooYn-Yn-1

4.常用符号

5.微分学基本公式

1.3.5.7.9.

y=cdy=O

IcosX

1.1+2+-+n=

n(n+l)2

2.12+22+…+n2=

nn+1(2n+l)

6

3.13+23+…+n3=(l+2+…n)24.a3±b3=a+b(a2Tab+b2)

5.xn-1=x-1(xn-1+xn-2+…+x+1)

6.xn-an=x-a(xn-1+axn-2+a2xn-3+“+a2x+an

-1)7.xn+an=x+a[x2k-1-ax2k-2+…+(a2k-2x-a2k

-1)]8.x-1=(++-+1)9.伯努利不等式(1+x)n>l+nx

n

1+xll+x2•••1+xn>l+xl+x2+-+XI1

10.|x-y|>|x-|y||

11.|xy|>xy

12.X+Xl+-+Xn>X-X1++Xn13.n!ln+1

n+ln

)2

1

14.....

24

132n-12n

n

n

a-In

16.1+a17.n-Ik

Ak

18.组合数公式Cn=

k!

n!

k!n-k!

kk-1

Cm+n+l-Cmk=Cm+n+n

n!

k

排列数公式An=n,n-1-n-k+1=

n-k!

19.z6-1=z+1z-1z2+z+1z2-z+120.z6+1=z2+1

(z4-z2+1)21.z4+1=z2++1(z2-+1)

1.记号n!!表示自然数的连乘积，这些自然数不超过n,

并且每两个数之间差2.例：7!!=1-3-5-78!!=2-4-6-8

dx

2.4.6.8.

y=xNdy=|ixg-ldxy=logaxdy=

logaex

y=axdy=axInadxy=sinxdy=cosxdxy=tanxdy=sec

2xdx=

dx

y=cosxdy=-sinxdxy=cotxdy=-csc2xdx=

Isinx

dx

y=secxdy=secxtanxdx

11+xlch2

x

10.y=cscxdy=-cscxcotxdx12.y=arccosxdy=14.y

=arccotxdy=-18.y=cthxdy=-

11+xlsh2x

11.y=arcsinxdy=13.y=arctanxdy=17.y=thxdy=

dxdx

15.y=shxdy=chxdx

dx

16.y=chxdy=shxdx

dx

6.不定积分表

1.Odx=c3.xjidx=5.

11+xlg+l

2.Idx=x+c4.dx=ln|x|+c.

x1

dx=arctanx+c

axIna

6.

=arcsinx+c

7.axdx=+c8.sinxdx=-cosx+c10.14.16

Isin2xIshxdxax+x2xdxa±x9.cosxdx=sinx+c11.

dx=tanx+c12.shxdx=chx+c

dx=-cthx+c=arctan+c,(aRO)

13.chxdx=shx+c15.2dx=thx+c

chx17.=lnll+c

a+x

18.20.

=±ln|a2±x2|+c

2

1

19.arcsin+c,(a>0)

a21.=ln|x++c

±+c

x2

22.dx=+arcsin+c,(a>0)

7.三角学公式sin20cos20

1.基本关系

1.sinO.csc0=13.tanO.cot0=15.sec20-tan20=17.tan

0=

sinOcos0

22.dx=±

2

x

a22

Inx++c

2.cosO.sec0=14.sin20+cos20=16.esc20-cot20=18.cot

0=

cosOsin0

2.两角和与差的三角函数公式

1.sina±p=sinacosp±cosasin03.tana±p=

tana±tanplTtanatanp

2.cosa±p=cosacos0干sinasin04.cota±p=

cotacotpTlcotp±cota

3.倍角公式

1.sin2a=2sinacosa=

2tanal+tana

1-tan2al+tana

2.cos2a=cos2a-sin2a=2cos2a-l=l-2sin2a=

3.tan2a=

2tanal-tana

=(

sinal+cosa

2.2

4.cot2a=

cot2a-12cota

5.sin3a=3sina-4sin3a1.3.

sin2=

2a

1-cosa

2

6.cos3a=4cos3a-3cosacos2=

2a

l+cosa

2

4.半角公式

=(

tan2=

2

al-cosal+cosal-cosa2

)sina

4.

cot2=

2

al+cosal-cosa

=(

l+cosa2

sina

=(

sina

1-cosa

)2

5.和差化积公式

1.2.3.4.5.6.7.

sina+sinp=2sinsina-sinp=2cos

a+p2a+p2

cossin

a-p2a-p2

cosa+cosp=2cos

a+p2

cos

a—02

cosa-cosp=-2sintana±tanp=±cota±cotp=±tana±cot

P=±

a+p2

sin

a—02

sin(a±p)sinasinp

cos(aTp)sinasinp

cos(a邛)cosasinp

6.积化和差公式

1.2.3.

sinasinp=-[cosa+p-cos(a-p)]cosacosp=[cosa+p

+cos(a-p)]sinacosp=[sina+p+sin(a-p)]

221121

7.双曲函数的基本关系

1.3.5.7.

cosh2t-sinh2t=1coth2t=l+sinhx=

2

Isinh2t

2.4.6.

1-tanh2t=

Icosht

sinh2x=2sinhxcoshx=2cosh2x-tanhxcoshx=

ex+e-x

2

ex-e-x

双曲余弦的反函数

x1+tanx

x=ln(y±y

8.万能公式

1.3.5.

sinx=tanx=secx=

2tan

2.4.6.

COSX=cotX=CSCX=

1-tan2x1+tanx1-tan2x2tan

2

x1-tanx

2tan

l+tan2x1-tanx

l+tan2x2tan

2

考研常用积分公式（4）

常用积分公式

(一)含有ax+b的积分(ar0)

dxllnax+b+CJl.ax+b=a

l|i+l(ax+b)+C(ax+b)dxa(|i+l)J2.=(|#-1)u

xlx(ax+b-bInax+b)+C2Jax+ba3.=

1F11x222(ax+b)-2b(ax+b)+bInax+b+Cx3IIfa

2J4.ax+b=L

dxlax+b-Jx(ax+b)bInx+C

5.=

dxlaax+b-+ln+CJx2(ax+b)bxb2x6.=

xlbx(Inax+b+)+C2J(ax+b)2aax+b7.=

2x21bJ(ax+b)2xa3(ax+b-2bInax+b-ax+b)+C

8.=

dxllax+b-In+CJx(ax+b)2b(ax+b)b2x9.=

的积分

CxlO

2(3ax-2b+Cx211

=15a

12

・J

x2222(15ax-12abx+8bCx3=105a

13

2x(ax-2bC=3a2

14

*

22x(3a2x2-4abx+8b2+C3=15a

f

15

.=+C(b>0)C(b16

・J

a-2b17

.bx=18

.a+x2=

22(三)含有x土a的积分

dxlxarctan+C22Jx+aaa19.=

x2n-3dxdx+J(x2+a2)n2(n-1)a2(x2+a2)n-12(n-1)

a2J(x2+a2)n-120.=

lx-adxIn+C22J21.x-a=2ax+a

2(四)含有ax+b(a>0)的积分

fdxJ

222.ax+b=x+C(b>0)+C(bxlxInax2+b+C

2J23.ax+b=2a

x2xbdx-xJ224.ax+b=aaax+b

lx2dxIn2+Cif2bax+b25.x(ax+b)=

dxladxJx2(ax2+b)-bx-bJax2+b26.=

ax2+bdxalln-+CJx3(ax2+b)2b222x2bx27.=

xIdxdx+J(ax2+b)22b(ax2+b)2bJax2+b28.=

2(五)含有ax+bx+c(a>0)的积分

2+C(bax2+bx+c=+C

x

30.Jax2+bx+cx1

=2aInax2+bx+c-bdx

2aJax2+bx+c

(a>0)的积分

31

.=arshx

a+

Cl=ln(x++C

32

+c

x33

.C

34

.J

x=+C

2

35

*

xa2

21n(x+C

2

x+ln(x++C36

.=(b2>4ac)

37

.J

1+Ca

38

*

+C=39

2a+ln(x++Cx2

40

x3422(2x+5aaln(x++Cx8=8Cx41

.J

42

•xj

4xa22(2x+aln(x++Cx8=8

43

*

aaIn+Cxx=

44

*

x+ln(x+C—

(a>0)的积分45

.J

xxarch+

Cllnx+Ca=x=46

.+C

47

xC48

+C

49

f

22xa21nx++C250

.x+lnx++C

51

♦J

laarccos+Cax52

.+C=

53

*

2aInx++Cx2

54

.J

x3422(2x-5aaInx++Cx8=8Cx55

.J

x56

.J

4xa22(2x-aInx++Cx88=

57

.J

aaarccos+Cxx=

58

*

x+lnx++C=

(a>0)的积分

59

.=arcsinx+Ca

60

.J

+Cx61

.=C

62

.J

x+C

63

I

2axxarcsin+C2a=22

x-arcsin64

.x+Ca

65

・J

la-In+Cax

66

*

+C=67

2axarcsin+Cx2a

68

♦J

x34x22(5a-2xaarcsin+Cx88a=Cx69

・J

=70

•xj

4xax22(2x-aarcsin+Cx88a=

71

*

x=a+C

72

*

xx-arcsin+Ca=

(a>0)的积分73

2ax+b++C

x74

.J

+2

2ax+b++C

75

2ax+b++C

76

.J

=+c

77

.J

2+Cx=

+C=

78

79

f

x(x-b(b-a=+C80

.x(x-b(b-aC=C81

.(a82

・J

(b-a)2Cx4(a(十一)含有三角函数的积分

sinxdx-cosx+CJ83.=

cosxdxsinx+CJ84.=

tanxdx-Incosx+CJ85.=

86.Jcotxdx=lnsinx+C

7txIntan(+)+CsecxdxInsecx+tanx+CJ4287.==

escxdxJ88.=

2secJxdxIntanx+CInescx-cotx+C2=89.=tanx

+C

=-cotx+C90.2cscJxdx

secxtanxdxsecx+CJ91.=

escxcotxdx-escx+CJ92.=

x1-sin2x+CsinxdxJ93.=242

x1+sin2x+CcosxdxJ2494.=2

In-In-1-sinxcosx+sinn-2xdxsinxdxJn95.J=nn

In-In-Icosxsinx+cosn-2xdxcosxdxJn96.J=nn

dxIcosxn-2dx-+n-1JnJn-297.sinx=n-Isinxn

-Isinx

dxIsinxn-2dx+n-1JnJn-298.cosx=n-Icosxn

-Icosx

Im-Im-In+lcosxsinx+cosm-2xsinnxdxcosxsinx

dxJm+n99.J=m+nmn

In-Imn-2cosm+lxsinn-lx+cosxsinxdxJm+n=

m+n-

llcos(a+b)x-cos(a-b)x+CsinaxcosbxdxJ2(a+b)

2(a-b)100.=-

llsin(a+b)x+sin(a-b)x+CsinaxsinbxdxJ2(a+b)

2(a-b)101.=-

llsin(a+b)x+sin(a-b)x+Ccosaxcosbxdx2(a+b)

f2(a-b)102.=

x+b+CdxJ

103.a+bsinxatan(a2>b2)

+CdxJ

104.a+bsinx=(a2xdx)+CJ

2105.a+bcosxdxJ

106.a+bcosx=+C(a2>b2)(a2dxlbarctan(tanx)+C

2222Jacosx+bsinxaba107.=

lbtanx+adxIn+C2222J2abbtanx-a108.acosx-b

sinx=

Hsinax-xcosax+Cxsinaxdx2Ja109.=a

1222xcosax+xsinax+cosax+Cxsinaxdx23Jaa

110.=a2-

llcosax+xsinax+Cxcosaxdx2Jaa111.=

1222xsinax+xcosax-sinax+Cxcosaxdx23faaa

112.=2

(十二)含有反三角函数的积分(其中a>0)

xxarcsinxxarcsin++CJ

a=a113.

x2a2xxxarcsindx(-)arcsin+CJ

a=24a114.

x3x12xxarcsinxarcsin+(x+2a2CJ

a=3a9115.2

xxarccosxxarccos-CJ

aa116.=

x2a2xxxarccosx(-)arccos-CJ

a=24a117.

x3x12x2xarccosdxarccos-(x+2aCJ

a=3a9118.2

xxa22arctandxxarctan-ln(a+x)+CJa=a2119.

x12xa2xarctandx(a+x)arctan-x+CJa=2a2120.

x3xa2a3xxarctandxarctan-x+ln(a2+x2)+CJa=

3a66121.2

(十三)含有指数函数的积分

lxa+CadxJina122.=x

laxe+CedxJa123.=ax

l(ax-l)eax+Cxedx2124.J=aax

125.naxxJedxInaxnn-laxxe-Jxedxa=a

xxlxa-a+C2xadxInaJ(Ina)126.=x

Inxnxa-xn-laxdxxadxjlna127.J=lnan