微积分公式

微積分公式 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  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) = -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 ( a x )= 22 1 xa   cos-1 ( a x )= tan-1 ( a x )= 22 xa a   cot-1 ( a x )= sec-1 ( a x )= 22 axx a   csc-1 (x/a)=  sin-1 x dx = x sin-1 x+ 21 x +C  cos-1 x dx = x cos-1 x- 21 x +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  sec-1 x dx = x sec-1 x- ln |x+ 12 x |+C  csc-1 x dx = x csc-1 x+ ln |x+ 12 x |+C sinh-1 ( a x )= ln (x+ 22 xa  ) xR cosh-1 ( a x )=ln (x+ 22 ax  ) x≧1 tanh-1 ( a x )= a2 1 ln ( xa xa   ) |x| <1 coth-1 ( a x )= a2 1 ln ( ax ax   ) |x| >1 sech (-1 a x )=ln( x 1 + 2 21 x x )0≦x≦1 )=ln( x 1 + 2 21 x xcsch-1 ( a x ) |x| >0 Dx sinh x = cosh x cosh x = sinh x anh x oth x  sinh x dx = cosh x + C  cosh x dx = sinh x + C C tanh x = sech2 x coth x = -csch2 x sech x = -sech x t csch x = -csch x c  tanh x dx = ln | cosh x |+  coth x dx = ln | sinh x | + C  sech x dx = -2tan-1 (e-x) + C  csch x dx = 2 ln | xe 21  xe1  | + C v  duv = uv =  udv +  vdu du osh2θ duv = ud + vdu → udv = uv -  v cos2θ-sin2θ=cos2θ cos2θ+ sin2θ=1 cosh2θ-sinh2θ=1 cosh2θ+sinh2θ=c D sinhx -1( a x )= 22 1 xa  cosh-1( a x )= 22 1 ax  tanh-1( a x )= 22 xa a   coth-1( a x )= sech-1( a x )= 2x2ax  csch-1(x/a)= a  2x a  2ax  h-1 x- sinh-1 x dx = x sin 2x + 1 C  cosh-1 x dx = x cosh-1 x- 12 x + C  tanh-1 x dx = x tanh-1 x+ ½ ln | 1-x2|+ C  coth-1 x dx = x coth-1 x- ½ ln | 1-x2|+ C  sech-1 x dx = x sech-1 x- sin-1 x + C  csch-1 x dx = x csch-1 x+ sinh-1 x + C 3θ →sin3θ= ¼ (3sinθ-sin3θ) →cos3θ=¼(3cosθ+cos3θ) sin x = sin 3θ=3sinθ-4sin cos3θ=4cos3θ-3cosθ j ee jxjx 2  cos x = 2 jxjx ee  sinh x = 2 xx ee  cosh x = 2 xx ee  正弦定理: sin a = sin b = sin c =2R 餘弦定理: a2=b2+c2-2bc cosα b2=a2+c2-2ac cosβ c2=a2+b2-2ab cosγ γ a b c αβ R sin (α β n α o± )=si cos β ± c s β co c cos β n α sin 2 sin α cos β = sin (α+β) + s ) 2 n sin (α+β - si 2 cos cos α-β + c 2 cos (α-β) - co sin α β = sin α - sin β = cos α β = (α+β) sin α sin + sin s (α±β)= os α i s β in (α-β + cos cos α si β = ) α β = cos ( ) n (α-β) cos α - cos β = os (α+β) tan (α±β)= sin α sin β = s (α+β) 2 sin ½(α+β) cos ½(α-β) 2 cos ½(α+β) sin ½(α-β) 2 cos ½(α+β) cos ½(α-β) -2 sin ½ ½(α-β)   tantan tantan  , cot (α±β)=   cotcot cot  cot ex=1+x+ !2 2x + !3 3x +…+ !n xn + … sin x = x- !3 3x + !5 5x - !7 7x +…+ )!12( )1( 12    n x nn + … cos x = 1- !2 2x + !4 4x - !6 6x +…+ )!2( )1( 2 n x nn + … ln (1+x) = x- 2 2x + 3 3x - 4 4x +…+ )!1( )1( 1    n xnn + … tan x = x--1 3 + 3x 5 - 5x 7 7x +…+ )12( n )1( xn 12 n + … (1+x)r =1+rx+ !2 )1( rr x2+ !3 x3+… -1