γ
Ra
b
β
α
c
sin x = e jx e jx cos x = e jx e jx
2j
2
sinh x = e x e x cosh x = e x e x
2
2
正弦定理: a = b = c =2R sin sin sin
餘弦定理: a2=b2+c2-2bc cosα
b2=a2+c2-2ac cosβ
sec-1 x dx = x sec-1 x- ln |x+
x2 1 |+C
coth-1 ( x )= 1 ln ( x a ) |x| >1 a 2a x a
csc-1 x dx = x csc-1 x+ ln |x+ x2 1 |+C sech-1( x )=ln( 1 + 1 x2 )0≦x≦1
coth x = -csch2 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 csch x dx = 2 ln | 1 ex | + C 1 e2x
1 a2 x2
1 x2 a2
sinh-1 x dx = x sinh-1 x- 1 x2 + C
sin 3θ=3sinθ-4sin3θ
cosh-1 x dx = x cosh-1 x- x2 1 + C cos3θ=4cos3θ-3cosθ
tanh-1 x dx = x tanh-1 x+ ½ ln | 1-x2|+ C →sin3θ= ¼ (3sinθ-sin3θ)
2 34
(n 1)!
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-1 x dx = x cos-1 x- 1 x2 +C tan-1 x dx = x tan-1 x-½ln (1+x2)+C cot-1 x dx = x cot-1 x+½ln (1+x2)+C
cosh-1 ( x )=ln (x+ x2 a2 ) x≧1 a
tanh-1 ( x )= 1 ln ( a x ) |x| <1 a 2a a x
a
x
x2
csch-1 ( x )=ln( 1 +
a
x
1 x2 x2
)
|x| >0
Dx sinh x = cosh x
sinh x dx = cosh x + C
cosh x = sinh x
cosh x dx = sinh x + C
tanh x = sech2 x
tanh x dx = ln | cosh x |+ C
ex=1+x+ x2 + x3 +…+ xn + …
2! 3!
n!
sin x = x- x3 + x5 - x7 +…+ (1)n x2n1 + …
3! 5! 7!
(2n 1)!
cos x = 1- x2 + x4 - x6 +…+ (1)n x2n + …
2! 4! 6!
(2n)!
ln (1+x) = x- x2 + x3 - x4 +…+ (1)n xn1 + …
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
微積分公式
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
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)
tanh-1( x )= a
a a2 x2
coth-1( x )= a
sech-1( x )=
a
a x a2 x2
csch-1(x/a)= a x a2 x2
coth-1 x dx = x coth-1 x- ½ ln | 1-x2|+ C →cos3θ=¼(3cosθ+cos3θ)
sech-1 x dx = x sech-1 x- sin-1 x + C csch-1 x dx = x csch-1 x+ sinh-1 x + 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
Dx sin-1 ( x )=
1
a
a2 x2
duv = udv + vdu duv = uv = udv + vdu → udv = uv - vdu
cos2θ-sin2θ=cos2θ
cos2θ+ sin2θ=1
cosh2θ-sinh2θ=1
cosh2θ+sinh2θ=cosh2θ
Dx sinh-1( x )= a
cosh-1( x )= a
cos-1 ( x )= a
tan-1
(
Байду номын сангаас
x a
)=
a a2 x2
cot-1 ( x )= a
sec-1 ( x )= a a x x2 a2
csc-1 (x/a)=
sin-1 x dx = x sin-1 x+ 1 x2 +C
sinh-1 ( x )= ln (x+ a2 x2 ) x R a