2024年5月12日发(作者：琦夏波)

微积分公式

D

x

sin x=cos x

cos x = -sin x

tan x = sec

2

x

cot x = -csc

2

x

sec x = sec x tan x

csc x = -csc x cot x

1

x

D

x

sin

-1

()=

22

a

ax

x

cos

-1

()=

a

x

a

tan

-1

()=

2

a

ax

2

sin x dx = -cos x + C

cos x dx = sin x + C

tan x dx = ln |sec x | + C

cot x dx = ln |sin x | + C

sec x dx = ln |sec x + tan x | + C

csc x dx = ln |csc x – cot x | + C

sin

-1

x dx = x sin

-1

x+

1x

2

+C

cos

-1

x dx = x cos

-1

x-

1x

2

+C

tan

-1

x dx = x tan

-1

x-ln (1+x

2

)+C

cot

-1

x dx = x cot

-1

x+ln (1+x

2

)+C

sec

-1

x dx = x sec

-1

x- ln |x+

x

2

1

|+C

sin

-1

(-x) = -sin

-1

x

cos

-1

(-x) = - cos

-1

x

tan

-1

(-x) = -tan

-1

x

cot

-1

(-x) = - cot

-1

x

sec

-1

(-x) = - sec

-1

x

csc

-1

(-x) = - csc

-1

x

x

sinh

-1

()= ln (x+

a

2

x

2

) x



R

a

x

cosh

-1

()=ln (x+

x

2

a

2

) x≧1

a

x1

ax

tanh

-1

()=ln () |x| <1

a2a

ax

x

cot

-1

()=

a

1

xa

-1

x

coth ()=ln () |x| >1

2

-1-1

a2a

xa

csc x dx = x csc x+ ln |x+

x1

|+C

x1

1x

2

sech()=ln(+)0≦x≦1

2

ax

x

-1

sec

-1

(

x

a

)=

22

a

xxa

csc

-1

(x/a)=

D

x

sinh x = cosh x

cosh x = sinh x

tanh x = sech

2

x

sinh x dx = cosh x + C

cosh x dx = sinh x + C

tanh x dx = ln | cosh x |+ C

x1

1x

2

csch ()=ln(+) |x| >0

ax

x

2

duv = udv + vdu

-1

duv = uv = udv + vdu

→ udv = uv - vdu

cos

2

θ-sin

2

θ=cos2θ

cos

2

θ+ sin

2

θ=1

cosh

2

θ-sinh

2

θ=1

cosh

2

θ+sinh

2

θ=cosh2θ

sin 3θ=3sinθ-4sin

3

θ

cos3θ=4cos

3

θ-3cosθ

→sin

3

θ= (3sinθ-sin3θ)

→cos

3

θ=(3cosθ+cos3θ)

coth x = -csch

2

x coth x dx = ln | sinh x | + C

sech x = -sech x tanh x sech x dx = -2tan

-1

(e

-x

) + C

csch x = -csch x coth x

1e

x

csch x dx = 2 ln || + C

2x

1e

1

x

D

x

sinh

-1

()=

sinh

-1

x dx = x sinh

-1

x-

1x

2

+ C

22

a

ax

x

cosh

-1

()=

a

1

xa

22

cosh

-1

x dx = x cosh

-1

x-

x

2

1

+ C

x

a

tanh

-1

()=

2

2

a

ax

-1

tanh

-1

x dx = x tanh

-1

x+ ln | 1-x

2

|+ C

e

jx

e

jx

e

jx

e

jx

sin x = cos x =

-1-12

2

2j

coth x dx = x coth x- ln | 1-x|+ C

e

x

e

x

e

x

e

x

sinh x = cosh x =

22

bc

a

正弦定理:

= ==2R

sin



sin



sin



sech

-1

x dx = x sech

-1

x- sin

-1

x + C

x

coth()=

a

csch

-1

x dx = x csch

-1

x+ sinh

-1

x + C

a

x

γ

sech

-1

()=

22

a

a

xax

R b

csch

-1

(x/a)=

a

xax

22

β

α

c

余弦定理: a

2

=b

2

+c

2

-2bc cosα

b

2

=a

2

+c

2

-2ac cosβ

c

2

=a

2

+b

2

-2ab cosγ

sin (α±β)=sin α cos β ± cos α sin β

cos (α±β)=cos α cos β



sin α sin β

2 sin α cos β = sin (α+β) + sin (α-β)

sin α + sin β = 2 sin (α+β) cos (α-β)

sin α - sin β = 2 cos (α+β) sin (α-β)

cos α + cos β = 2 cos (α+β) cos (α-β)

2 cos α sin β = sin (α+β) - sin (α-β)

2 cos α cos β = cos (α-β) + cos (α+β)

2 sin α sin β = cos (α-β) - cos (α+β)

2n

3

xx

x

e

x

=1+x+++…++ …

2!

3!

n!

cos α - cos β = -2 sin (α+β) sin (α-β)

tan



tan



cot



cot



tan (α±β)=, cot (α±β)=

tan



tan



cot



cot





1

= n

i1

n

n

x

3

x

5

x

7

(1)

n

x

2n1

sin x = x-+-+…++ …

3!5!

7!

(2n1)!

x

2

x

4

x

6

(1)

n

x

2n

cos x = 1-+-+…++ …

2!4!6!

(2n)!

x

2

x

3

x

4

(1)

n

x

n1

ln (1+x) = x-+-+…++ …

2

3

4

(n1)!



i

= n (n+1)

i1

n



i

2

=

i1

n

1

n (n+1)(2n+1)

6



i

i1

3

= [n (n+1)]

2

x

3

x

5

x

7

(1)

n

x

2n1

tan x = x-+-+…++ …

35

7

(2n1)

-1

Γ(x) =



t

0



x-1-t

e dt = 2



t

0



2x-1

t

2

e

dt =





0

1

(ln)

x-1

dt

t

(1+x)

r

=1+rx+



1

r(r1)

2

r(r1)(r2)

3

x+x+… -1

β(m, n) =



x

m-1

(1-x)

n-1

dx=2

2

sin

2m-1

x cos

2n-1

x dx



0

0

2!

3!

=

希腊字母 (Greek Alphabets)

大写

Α

Β

Γ

Δ

Ε

Ζ

Η

Θ

小写





0

x

m1

dx

(1x)

mn

α

β

γ

δ

ε

ζ

η

θ

读音

alpha

beta

gamma

delta

epsilon

zeta

eta

theta

大写

Ι

Κ

Λ

Μ

Ν

Ξ

Ο

Π

小写

ι

κ

λ

μ

ν

ξ

ο

π

读音

iota

kappa

lambda

mu

nu

xi

omicron

pi

大写

Ρ

Σ

Τ

Υ

Φ

Χ

Ψ

Ω

小写

ρ

σ,

sigma

tau

τ

υ

upsilon

phi

φ

khi

χ

psi

ψ

ω

omega

读音

rho

倒数关系: sinθcscθ=1; tanθcotθ=1; cosθsecθ=1

商数关系: tanθ=

sin



cos



; cotθ=

cos



sin



平方关系: cos

2

θ+ sin

2

θ=1; tan

2

θ+ 1= sec

2

θ; 1+ cot

2

θ= csc

2

θ

順位高

; 顺位高d 顺位低 ;

順位低

0* =

1

10

* = = 0* =

00



0

0

=

e

0()

;



0

=

e

0

;

1



=

e

0

算术平均数(Arithmetic mean)

顺位一: 对数; 反三角(反双曲)

顺位二: 多项函数; 幂函数

顺位三: 指数; 三角(双曲)

中位数(Median)

众数(Mode)

几何平均数(Geometric mean)

调和平均数(Harmonic mean)

平均差(Average Deviatoin)

取排序后中间的那位数字

次数出现最多的数值

变异数(Variance)



(X

1

n

i

X)

2

n

or



(X

1

n

i

X)

2

n1

标准差(Standard Deviation)



(X

1

n

i

X)

2

n

分配

Discrete

Uniform

Continuous

Uniform

Bernoulli

Binomial

Negative

Binomial

Multinomial

机率函数f(x)

or



(X

1

n

i

X)

2

变异数V(x)

1

2

(n+1)

12

1

(b-a)

2

12

n1

期望值E(x)

1

(n+1)

2

1

(a+b)

2

动差母函数

m(t)

p

x

q

1-x

(x=0, 1)



n



xn-x





x





pq





kx1



kx





pq

x



f(x

1

, x

2

, …,

x

m-1

)=

n!

xxx

p

1

1

p

2

2

...p

m

m

x

1

!x

2

!...x

m

!

p

np

pq

npq

q+pe

t

(q+ pe

t

)

n

np

i

np

i

(1-p

i

)

三项

(p

1

e

t1

+

p

2

e

t2

+ p

3

)

n

Geometric

Hypergeometric

Poisson

Normal

Beta

Gamma

pq

x-1



k



n





N





Nn





k





n





N1





N



λ

μ

λ

σ

2

Exponent

Chi-Squaredχ

2

=f(χ

2

E(χ)=n

1



n







2



2



n

2

2

V(χ)=2n

2

)

=

(



2

)

n

1

2

e





2

2

Weibull

1 000 000 000 000 000 000 000 000 10

24

yotta Y

1 000 000 000 000 000 000 000 10

21

zetta Z

1 000 000 000 000 000 000 10

18

exa E

1 000 000 000 000 000 10

15

peta P

1 000 000 000 000 10

12

tera T 兆

1 000 000 000 10

9

giga G 十亿

1 000 000 10

6

mega M 百万

1 000 10

3

kilo K 千

100 102 hecto H 百

10 101 deca D 十

10

-1

deci d 分，十分之一

10

-2

centi c 厘（或写作「厘」），百分之一

10

-3

milli m 毫，千分之一

001 10

-6

micro 微，百万分之一

000 001 10

-9

nano n 奈，十亿分之一

000 000 001 10

-12

pico p 皮，兆分之一

000 000 000 001 10

-15

femto f 飞（或作「费」），千兆分之一

000 000 000 000 001 10

-18

atto a 阿

000 000 000 000 000 001 10

-21

zepto z

000 000 000 000 000 000 001 10

-24

yocto y

2024年5月12日发(作者：琦夏波)

微积分公式

D

x

sin x=cos x

cos x = -sin x

tan x = sec

2

x

cot x = -csc

2

x

sec x = sec x tan x

csc x = -csc x cot x

1

x

D

x

sin

-1

()=

22

a

ax

x

cos

-1

()=

a

x

a

tan

-1

()=

2

a

ax

2

sin x dx = -cos x + C

cos x dx = sin x + C

tan x dx = ln |sec x | + C

cot x dx = ln |sin x | + C

sec x dx = ln |sec x + tan x | + C

csc x dx = ln |csc x – cot x | + C

sin

-1

x dx = x sin

-1

x+

1x

2

+C

cos

-1

x dx = x cos

-1

x-

1x

2

+C

tan

-1

x dx = x tan

-1

x-ln (1+x

2

)+C

cot

-1

x dx = x cot

-1

x+ln (1+x

2

)+C

sec

-1

x dx = x sec

-1

x- ln |x+

x

2

1

|+C

sin

-1

(-x) = -sin

-1

x

cos

-1

(-x) = - cos

-1

x

tan

-1

(-x) = -tan

-1

x

cot

-1

(-x) = - cot

-1

x

sec

-1

(-x) = - sec

-1

x

csc

-1

(-x) = - csc

-1

x

x

sinh

-1

()= ln (x+

a

2

x

2

) x



R

a

x

cosh

-1

()=ln (x+

x

2

a

2

) x≧1

a

x1

ax

tanh

-1

()=ln () |x| <1

a2a

ax

x

cot

-1

()=

a

1

xa

-1

x

coth ()=ln () |x| >1

2

-1-1

a2a

xa

csc x dx = x csc x+ ln |x+

x1

|+C

x1

1x

2

sech()=ln(+)0≦x≦1

2

ax

x

-1

sec

-1

(

x

a

)=

22

a

xxa

csc

-1

(x/a)=

D

x

sinh x = cosh x

cosh x = sinh x

tanh x = sech

2

x

sinh x dx = cosh x + C

cosh x dx = sinh x + C

tanh x dx = ln | cosh x |+ C

x1

1x

2

csch ()=ln(+) |x| >0

ax

x

2

duv = udv + vdu

-1

duv = uv = udv + vdu

→ udv = uv - vdu

cos

2

θ-sin

2

θ=cos2θ

cos

2

θ+ sin

2

θ=1

cosh

2

θ-sinh

2

θ=1

cosh

2

θ+sinh

2

θ=cosh2θ

sin 3θ=3sinθ-4sin

3

θ

cos3θ=4cos

3

θ-3cosθ

→sin

3

θ= (3sinθ-sin3θ)

→cos

3

θ=(3cosθ+cos3θ)

coth x = -csch

2

x coth x dx = ln | sinh x | + C

sech x = -sech x tanh x sech x dx = -2tan

-1

(e

-x

) + C

csch x = -csch x coth x

1e

x

csch x dx = 2 ln || + C

2x

1e

1

x

D

x

sinh

-1

()=

sinh

-1

x dx = x sinh

-1

x-

1x

2

+ C

22

a

ax

x

cosh

-1

()=

a

1

xa

22

cosh

-1

x dx = x cosh

-1

x-

x

2

1

+ C

x

a

tanh

-1

()=

2

2

a

ax

-1

tanh

-1

x dx = x tanh

-1

x+ ln | 1-x

2

|+ C

e

jx

e

jx

e

jx

e

jx

sin x = cos x =

-1-12

2

2j

coth x dx = x coth x- ln | 1-x|+ C

e

x

e

x

e

x

e

x

sinh x = cosh x =

22

bc

a

正弦定理:

= ==2R

sin



sin



sin



sech

-1

x dx = x sech

-1

x- sin

-1

x + C

x

coth()=

a

csch

-1

x dx = x csch

-1

x+ sinh

-1

x + C

a

x

γ

sech

-1

()=

22

a

a

xax

R b

csch

-1

(x/a)=

a

xax

22

β

α

c

余弦定理: a

2

=b

2

+c

2

-2bc cosα

b

2

=a

2

+c

2

-2ac cosβ

c

2

=a

2

+b

2

-2ab cosγ

sin (α±β)=sin α cos β ± cos α sin β

cos (α±β)=cos α cos β



sin α sin β

2 sin α cos β = sin (α+β) + sin (α-β)

sin α + sin β = 2 sin (α+β) cos (α-β)

sin α - sin β = 2 cos (α+β) sin (α-β)

cos α + cos β = 2 cos (α+β) cos (α-β)

2 cos α sin β = sin (α+β) - sin (α-β)

2 cos α cos β = cos (α-β) + cos (α+β)

2 sin α sin β = cos (α-β) - cos (α+β)

2n

3

xx

x

e

x

=1+x+++…++ …

2!

3!

n!

cos α - cos β = -2 sin (α+β) sin (α-β)

tan



tan



cot



cot



tan (α±β)=, cot (α±β)=

tan



tan



cot



cot





1

= n

i1

n

n

x

3

x

5

x

7

(1)

n

x

2n1

sin x = x-+-+…++ …

3!5!

7!

(2n1)!

x

2

x

4

x

6

(1)

n

x

2n

cos x = 1-+-+…++ …

2!4!6!

(2n)!

x

2

x

3

x

4

(1)

n

x

n1

ln (1+x) = x-+-+…++ …

2

3

4

(n1)!



i

= n (n+1)

i1

n



i

2

=

i1

n

1

n (n+1)(2n+1)

6



i

i1

3

= [n (n+1)]

2

x

3

x

5

x

7

(1)

n

x

2n1

tan x = x-+-+…++ …

35

7

(2n1)

-1

Γ(x) =



t

0



x-1-t

e dt = 2



t

0



2x-1

t

2

e

dt =





0

1

(ln)

x-1

dt

t

(1+x)

r

=1+rx+



1

r(r1)

2

r(r1)(r2)

3

x+x+… -1

β(m, n) =



x

m-1

(1-x)

n-1

dx=2

2

sin

2m-1

x cos

2n-1

x dx



0

0

2!

3!

=

希腊字母 (Greek Alphabets)

大写

Α

Β

Γ

Δ

Ε

Ζ

Η

Θ

小写





0

x

m1

dx

(1x)

mn

α

β

γ

δ

ε

ζ

η

θ

读音

alpha

beta

gamma

delta

epsilon

zeta

eta

theta

大写

Ι

Κ

Λ

Μ

Ν

Ξ

Ο

Π

小写

ι

κ

λ

μ

ν

ξ

ο

π

读音

iota

kappa

lambda

mu

nu

xi

omicron

pi

大写

Ρ

Σ

Τ

Υ

Φ

Χ

Ψ

Ω

小写

ρ

σ,

sigma

tau

τ

υ

upsilon

phi

φ

khi

χ

psi

ψ

ω

omega

读音

rho

倒数关系: sinθcscθ=1; tanθcotθ=1; cosθsecθ=1

商数关系: tanθ=

sin



cos



; cotθ=

cos



sin



平方关系: cos

2

θ+ sin

2

θ=1; tan

2

θ+ 1= sec

2

θ; 1+ cot

2

θ= csc

2

θ

順位高

; 顺位高d 顺位低 ;

順位低

0* =

1

10

* = = 0* =

00



0

0

=

e

0()

;



0

=

e

0

;

1



=

e

0

算术平均数(Arithmetic mean)

顺位一: 对数; 反三角(反双曲)

顺位二: 多项函数; 幂函数

顺位三: 指数; 三角(双曲)

中位数(Median)

众数(Mode)

几何平均数(Geometric mean)

调和平均数(Harmonic mean)

平均差(Average Deviatoin)

取排序后中间的那位数字

次数出现最多的数值

变异数(Variance)



(X

1

n

i

X)

2

n

or



(X

1

n

i

X)

2

n1

标准差(Standard Deviation)



(X

1

n

i

X)

2

n

分配

Discrete

Uniform

Continuous

Uniform

Bernoulli

Binomial

Negative

Binomial

Multinomial

机率函数f(x)

or



(X

1

n

i

X)

2

变异数V(x)

1

2

(n+1)

12

1

(b-a)

2

12

n1

期望值E(x)

1

(n+1)

2

1

(a+b)

2

动差母函数

m(t)

p

x

q

1-x

(x=0, 1)



n



xn-x





x





pq





kx1



kx





pq

x



f(x

1

, x

2

, …,

x

m-1

)=

n!

xxx

p

1

1

p

2

2

...p

m

m

x

1

!x

2

!...x

m

!

p

np

pq

npq

q+pe

t

(q+ pe

t

)

n

np

i

np

i

(1-p

i

)

三项

(p

1

e

t1

+

p

2

e

t2

+ p

3

)

n

Geometric

Hypergeometric

Poisson

Normal

Beta

Gamma

pq

x-1



k



n





N





Nn





k





n





N1





N



λ

μ

λ

σ

2

Exponent

Chi-Squaredχ

2

=f(χ

2

E(χ)=n

1



n







2



2



n

2

2

V(χ)=2n

2

)

=

(



2

)

n

1

2

e





2

2

Weibull

1 000 000 000 000 000 000 000 000 10

24

yotta Y

1 000 000 000 000 000 000 000 10

21

zetta Z

1 000 000 000 000 000 000 10

18

exa E

1 000 000 000 000 000 10

15

peta P

1 000 000 000 000 10

12

tera T 兆

1 000 000 000 10

9

giga G 十亿

1 000 000 10

6

mega M 百万

1 000 10

3

kilo K 千

100 102 hecto H 百

10 101 deca D 十

10

-1

deci d 分，十分之一

10

-2

centi c 厘（或写作「厘」），百分之一

10

-3

milli m 毫，千分之一

001 10

-6

micro 微，百万分之一

000 001 10

-9

nano n 奈，十亿分之一

000 000 001 10

-12

pico p 皮，兆分之一

000 000 000 001 10

-15

femto f 飞（或作「费」），千兆分之一

000 000 000 000 001 10

-18

atto a 阿

000 000 000 000 000 001 10

-21

zepto z

000 000 000 000 000 000 001 10

-24

yocto y