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# 高中数学微积分公式大全

Dx sin x=cos x cos x = -sin x tan x = sec2 x cot x = -csc2 x sec x = sec x tan x csc x = -csc x cot x
x Dx sin-1 ( )= a

? 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

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

?1 a2 ? x2

x cos-1 ( )= a x ?a tan-1 ( )= 2 a a ? x2 x cot-1 ( )= a

sec-1 (

x ?a )= a x x2 ? a2

x sinh-1 ( )= ln (x+ a 2 ? x 2 ) x ? R a -1 -1 2 x ? cos x dx = x cos x- 1 ? x +C cosh-1 ( )=ln (x+ x 2 ? a 2 ) x≧1 a ? tan-1 x dx = x tan-1 x-?ln (1+x2)+C x 1 a?x tanh-1 ( )= ln ( ) |x| <1 -1 -1 2 a 2a a?x ? cot x dx = x cot x+?ln (1+x )+C 1 x?a -1 x ln ( ) |x| >1 ? sec-1 x dx = x sec-1 x- ln |x+ x 2 ? 1 |+C coth ( )= a 2a x?a ? csc-1 x dx = x csc-1 x+ ln |x+ x 2 ? 1 |+C x ?1 1 ? x2 sech-1( )=ln( + )0≦x≦1 a x x2

csc-1 (x/a)= Dx sinh x = cosh x cosh x = sinh x tanh x = sech2 x coth x = -csch2 x sech x = -sech x tanh x csch x = -csch x coth x ? sinh x dx = cosh x + C ? cosh x dx = sinh x + C ? tanh x dx = ln | cosh x |+ C ? coth x dx = ln | sinh x | + C ? sech x dx = -2tan-1 (e-x) + C ? csch x dx = 2 ln |

csch-1 (

x 1 1 ? x2 )=ln( + ) |x| >0 a x x2

duv = udv + vdu ? duv = uv = ? udv + ? vdu →? udv = uv - ? vdu cos2?-sin2?=cos2? cos2?+ sin2?=1 cosh2?-sinh2?=1 cosh2?+sinh2?=cosh2?

1 ? e? x 1 ? e ?2 x

|+C

x Dx sinh-1( )= a x cosh-1( )= a

1 a2 ? x2

? sinh-1 x dx = x sinh-1 x- 1 ? x 2 + C ? cosh-1 x dx = x cosh-1 x- x 2 ? 1 + C

sin 3?=3sin?-4sin3? cos3?=4cos3?-3cos?

1 x2 ? a2

? tanh-1 x dx = x tanh-1 x+ ? ln | 1-x2|+ C →sin3?= ? (3sin?-sin3?) ? coth-1 x dx = x coth-1 x- ? ln | 1-x2|+ C →cos3?=?(3cos?+cos3?) ? sech-1 x dx = x sech-1 x- sin-1 x + C ? csch-1 x dx = x csch-1 x+ sinh-1 x + C R a
β γ α

x ?a tanh-1( )= 2 a a ? x2 x coth-1( )= a

sin x =

sech-1(

x ?a )= a x a2 ? x2

e jx ? e ? jx e jx ? e ? jx cos x = 2 2j e x ? e?x e x ? e?x cosh x = 2 2
a b c = = =2R sin ? sin ? sin ?

b

sinh x =

csch-1(x/a)=

?a x a2 ? x2

c

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) 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)

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) cos α - cos β = -2 sin ?(α+β) sin ?(α-β) tan (α±β)=
tan? ? tan ? ? cot? cot ? , cot (α±β)= ? tan? tan ? cot? ? cot ?

ex=1+x+

x2 x3 xn + +…+ +… 2! 3! n!

?1 = n
i ?1 n

n

x3 x5 x7 ( ?1) n x 2 n ?1 sin x = x- + - +…+ +… 3! 5! 7! (2n ? 1)!

?i =
i ?1 n

?n (n+1)
1 n (n+1)(2n+1) 6

cos x = 1-

x2 x4 x6 ( ?1) n x 2 n + +…+ +… 2! 4! 6! (2n)!

?i
i ?1

2

=

ln (1+x) = x-

x2 x3 x4 ( ?1) n x n ?1 + - +…+ +… 2 3 4 (n ? 1)!

?i
i ?1

n

3

= [?n (n+1)]2
x-1 -t 2x-1 ? t ? t e dt = 2 ? t e dt =
2

x3 x5 x7 ( ?1) n x 2 n ?1 tan x = x- + - +…+ +… 3 5 7 (2n ? 1)
-1

Γ(x) =

?

?

0

0

?

?

0

1 (ln ) x-1 dt t

(1+x)r =1+rx+

r ( r ? 1) 2 r ( r ? 1)( r ? 2) 3 x+ x +… -1<x<1 2! 3!

?(m, n) = ? x m-1(1-x)n-1 dx=2 ? 2 sin 2m-1x cos2n-1x dx
0

1

?

0

=

?

?

0

x m?1 dx (1 ? x ) m?n

Ω ? ? ? ? ? ? ?

α β γ δ ε ζ η θ

ι

κ, ? sigma λ μ ν ξ ο ω
tau upsilon phi khi psi omega

sin ? cos ? ; cot?= cos ? sin ?

0*? =

1 0 1 ? *? = = 0* = 0 0 ? ?

0 0 = e 0 ( ?? ) ; ? 0 = e 0?? ; 1? = e 0??

X ?

X 1 ? X 2 ? ... ? X n n

G ? n X 1 ? X 2 ? ...? X n
H? 1 1 1 1 1 ( ? ? ... ? ) n x1 x2 xn

?| X
1

n

i

?X |

n

?(X
1

n

i

? X )2

?(X
or
1

n

i

? X )2

n

n ?1

? ( X i ? X )2
1

n

?(X
or
1

n

i

? X )2

n

n ?1

1 n
1 b?a

1 (n+1) 2
1 (a+b) 2

1 2 (n +1) 12
1 (b-a)2 12

1 e t (1 ? e nt ) n 1 ? et

e bt ? e at (b ? a)t

pxq1-x(x=0, 1)
? n ? x n-x ? ?p q ? x? ? ?

p np

pq npq

q+pet (q+ pet)n

Negative Binomial Multinomial

? k ? x ? 1? k x ? ?p q ? x ? ? ?
f(x1, x2, …, xm-1)=
n! x x x p1 1 p 2 2 ...p m m x1! x2 !...xm !

kq p

kq p2

pk (1 ? qet ) k

npi

npi(1-pi)

Geometric

pqx-1
? k ?? N ? k ? ? ?? ? x ?? n ? x ? ? ? ?? ? ?N? ? ? ?n? ? ?
e ?? ?x x!

1 p

q p2
? N ?n? ? k ? ? ?n? ? ? N ?1 ? ? N ?

pet 1 ? qet

Hypergeometric

?k? n? ? ?N?

Poisson

γ
δ

γ
κ2

e ? ? ( e ?1)
t

Normal

1 2??

e

?

1 x?? 2 ( ) 2 ?

e
? ???
? ?
1

? t? ? 2 t2

1 2

Beta

1 x? ?1 (1 ? x) ? ?1 B(? , ? )

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

Gamma

? (?x)? ?1 e ??x ?(? )

? ?2
1 ?2

? ? ? ? ? ?? ?t ?

??

Exponent Chi-Squaredξ2 =f(ξ2)

? e ? ?x
n ?1 2 ?? 2 2

?

? ? ?t

E(ξ2)=n
n 2

V(ξ2)=2n

(1 ? 2t )

?

n 2

=

1 ?n? ?? ?2 ? 2?

(? 2 )

e

Weibull

1

?

e

?

x??

?

? ?2 ? ? 1 ?? ?1 ? ? ? ? ?? ? 1? ? ? ??? ? 1? ? ? 2 ? ? 1?? ? ? ?? ?? ? ? ?? ?
1

2

?

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