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2024年3月21日发(作者：list的读音)
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
ax
x
cos
-1
()=
a
xa
tan
-1
()=
2
aax
2
x
cot
-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+
1x
2
+C
cos
-1
x dx = x cos
-1
x-
1x
2
+C
tan
-1
x dx = x tan
-1
x-½ln (1+x
2
)+C
cot
-1
x dx = x cot
-1
x+½ln (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
x1ax
tanh
-1
()=ln () |x| <1
a2aax
1xa
-1
x
coth ()=ln () |x| >1
2
-1-1
a2axa
csc x dx = x csc x+ ln |x+
x1
|+C
x1
1x
2
sech()=ln(+)0≦x≦1
2
ax
x
-1
x
a
sec
-1
()=
a
xx
2
a
2
csc
-1
(x/a)=
D
x
sinh x = cosh x
cosh x = sinh x
sinh x dx = cosh x + C
cosh x dx = sinh x + C
x1
1x
2
csch ()=ln(+) |x| >0
2
ax
x
duv = udv + vdu
-1
tanh x = sech
2
x
tanh x dx = ln | cosh x |+ C
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
1e
x
csch x dx = 2 ln || + C
2x
1e
1
x
D
x
sinh
-1
()=
sinh
-1
x dx = x sinh
-1
x-
1x
2
+ C
22
a
ax
x
cosh
-1
()=
a
-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= ¼ (3sinθ-sin3θ)
→cos
3
θ=¼(3cosθ+cos3θ)
1
xa
22
cosh
-1
x dx = x cosh
-1
x-
x
2
1
+ C
tanh
-1
x dx = x tanh
-1
x+ ½ ln | 1-x
2
|+ C
xa
tanh()=
2
aax
2
-1
e
jx
e
jx
e
jx
e
jx
sin x = cos x =
2j
2
coth
-1
x dx = x coth
-1
x- ½ ln | 1-x
2
|+ C
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
xax
R b
csch
-1
(x/a)=
e
x
e
x
e
x
e
x
sinh x = cosh x =
22
bc
a
正弦定理:= ==2R
sin
sin
sin
a
xax
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)
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)
x
2
x
3
x
n
e=1+x+++…++ …
2!
3!
n!
x
sin α + sin β = 2 sin ½(α+β) cos ½(α-β)
sin α - sin β = 2 cos ½(α+β) sin ½(α-β)
cos α + cos β = 2 cos ½(α+β) cos ½(α-β)
cos α - cos β = -2 sin ½(α+β) sin ½(α-β)
tan (α±β)=
tan
tan
cot
cot
, cot (α±β)=
tan
tan
cot
cot
1
= n
i1
n
n
(1)
n
x
2n1
x
3
x
5
x
7
sin x = x-+-+…++ …
(2n1)!
3!5!
7!
(1)
n
x
2n
x
2
x
4
x
6
cos x = 1-+-+…++ …
(2n)!
2!4!6!
(1)
n
x
n1
x
2
x
3
x
4
ln (1+x) = x-+-+…++ …
(n1)!
2
3
4
(1)
n
x
2n1
x
3
x
5
x
7
tan x = x-+-+…++ …
(2n1)
35
7
-1
r
i
= ½n (n+1)
i1
n
i
2
=
i1
n
1
n (n+1)(2n+1)
6
i
i1
3
= [½n (n+1)]
2
x-1-t2x-1
t
tt
e dt = 2
e
dt =
00
2
Γ(x) =
0
1
(ln)
x-1
dt
t
1
r(r1)
2
r(r1)(r2)
3
m-1n-1
(1+x)=1+rx+x+x+… -1 β(m, n) = x (1-x) dx=2 2 sin 2m-1 x cos 2n-1 x dx 0 0 2! 3! = 希腊字母 大写 Α Β Γ Δ Ε Ζ Η Θ 小写 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 0 x m1 dx mn (1x) 小写 ρ σ, ς τ υ φ χ ψ ω 读音 rho sigma tau upsilon phi khi psi omega 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* = 110 * = = 0* = 00 0 0 = e 0() ; 0 = e 0 ; 1 = e 0 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)
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