微积分公式大全上课讲义.doc

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微积分公式

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
ò sin x dx = -cos x + C
ò cos x dx = sin x + C
ò tan x dx = ln |sec x | + C
ò cot x dx = ln |sin x | + C
ò sec x dx = ln |sec x + tan x | + C
ò csc x dx = ln |csc x – cot x | + C
sin-1(-x) = -sin-1 x
cos-1(-x) = p - cos-1 x
tan-1(-x) = -tan-1 x
cot-1(-x) = p - cot-1 x
sec-1(-x) = p - sec-1 x
csc-1(-x) = - csc-1 x
Dx sin-1 ()=
cos-1 ()=
tan-1 ()=
cot-1 ()=
sec-1 ()=
csc-1 (x/a)=
ò sin-1 x dx = x sin-1 x++C
ò cos-1 x dx = x cos-1 x-+C
ò tan-1 x dx = x tan-1 x-½ln (1+x2)+C
ò cot-1 x dx = x cot-1 x+½ln (1+x2)+C
ò sec-1 x dx = x sec-1 x- ln |x+|+C
ò csc-1 x dx = x csc-1 x+ ln |x+|+C

sinh-1 ()= ln (x+) xR
cosh-1 ()=ln (x+) x≧1
tanh-1 ()=ln () |x| <1
coth-1 ()=ln () |x| >1
sech-1()=ln(+)0≦x≦1
csch-1 ()=ln(+) |x| >0
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
ò sinh x dx = cosh x + C
ò cosh x dx = sinh x + C
ò tanh x dx = ln | cosh x |+ C
ò coth x dx = ln | sinh x | + C
ò sech x dx = -2tan-1 (e-x) + C
ò csch x dx = 2 ln || + C
duv = udv + vdu
ò duv = uv = ò udv + ò vdu
→ò udv = uv - ò vdu
cos2θ-sin2θ=cos2θ
cos2θ+ sin2θ=1
cosh2θ-sinh2θ=1
cosh2θ+sinh2θ=cosh2θ
Dx sinh-1()=
cosh-1()=
tanh-1()=
coth-1()=
sech-1()=
csch-1(x/a)=
ò sinh-1 x dx = x sinh-1 x-+ C
ò cosh-1 x dx = x cosh-1 x-+ C
ò tanh-1 x dx = x tanh-1 x+ ½ ln | 1-x2|+ C
ò coth-1 x dx = x coth-1 x- ½ ln | 1-x2|+ C
ò sech-1 x dx = x sech-1 x- sin-1 x + C
ò csch-1 x dx = x csch-1 x+ sinh-1 x + C
a
γ

R
α
β
c
b

sin 3θ=3sinθ-4sin3θ
cos3θ=4cos3θ-3cosθ
→sin3θ= ¼ (3sinθ-sin3θ)
→cos3θ=¼(3cosθ+cos3θ)
sin x = cos x =
sinh x = cosh x =
正弦定理:= ==2R
余弦定理: a2=b2+c2-2bc cosα
b2=a2+c2-2ac cosβ
c2=a2+b2-2ab 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)
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)=, cot (α±β)=
ex=1+x+++…++ …
sin x = x-+-+…++ …
cos x = 1-+-+…++ …
ln (1+x) = x-+-+…++ …
tan-1 x = x-+-+…++ …
(1+x)r =1+rx+x2+x3+… -1<x<1
= n
= ½n (n+1)
= n (n+1)(2n+1)
= [½n (n+1)]2
Γ(x) = x-1e-t dt = 22x-1dt = x-1 dt
β(m, n) =m-1(1-x)n-1 dx=22m-1x cos2n-1x dx = dx
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α
alpha
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ι
iota
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ρ
rho
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β
beta
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kappa
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sigma
Γ
γ
gamma
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lambda
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τ
tau
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δ
delta
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μ
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υ
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khi
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ο
omicron
Ψ
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θ
theta
Π
π
pi
Ω
ω
omega

倒数关系: sinθcscθ=1; tanθcotθ=1; cosθsecθ=1
商数关系: tanθ= ; cotθ=
平方关系: cos2θ+ sin2θ=1; tan2θ+ 1= sec2θ; 1+ cot2θ= csc2θ

; ò 顺位高d 顺位低 ;
0*¥ = *¥ = = 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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