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微积分公式

Dx sin x=cos x cos x = -sin 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 xsinh-1 ()= ln (x+a2?x2) x?R axcosh-1 ()=ln (x+x2?a2) x≧1 ax1a?xtanh-1 ()=ln () |x| <1 a2aa?x tan x = sec2 x cot x = -csc2 x sec x = sec x tan x csc x = -csc x cot x ?1x-1-12Dx sin-1 ()= ? sin x dx = x sin x+1?x+C aa2?x2? cos-1 x dx = x cos-1 x-1?x2+C x cos-1 ()= a? tan-1 x dx = x tan-1 x-?ln (1+x2)+C x?atan-1 ()=2 2aa?xxcot-1 ()= a? cot-1 x dx = x cot-1 x+?ln (1+x2)+C ? sec-1 x dx = x sec-1 x- ln |x+x2?1|+C 1x?a-1xcoth ()=ln () |x| >1 -1-12a2ax?a? csc x dx = x csc x+ ln |x+x?1|+C x?11?x2sech()=ln(+)0≦x≦1 2axx-1sec-1 ( x?a)= axx2?a2csc-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 |x11?x2csch ()=ln(+) |x| >0 2axxduv = udv + vdu -11?e?x1?e?2x| + C ? duv = uv = ? udv + ? vdu →? udv = uv - ? vdu cos2θ-sin2θ=cos2θ cos2θ+ sin2θ=1 cosh2θ-sinh2θ=1 cosh2θ+sinh2θ=cosh2θ

xDx sinh()= a-11a?x1x?a222 ? sinh-1 x dx = x sinh-1 x-1?x2+ C ? cosh-1 x dx = x cosh-1 x-x2?1+ C sin 3θ=3sinθ-4sin3θ cos3θ=4cos3θ-3cosθ →sin3θ= ? (3sinθ-sin3θ) →cos3θ=?(3cosθ+cos3θ) xcosh-1()= a-12x?atanh()= 2 2aa?x-1? tanh-1 x dx = x tanh-1 x+ ? ln | 1-x2|+ C ejx?e?jxejx?e?jxsin x = cos x = -1-122j2? coth x dx = x coth x- ? ln | 1-x|+ C ex?e?xex?e?xsinh x = cosh x = 22bca正弦定理:= ==2R sin?sin?sin?? sech-1 x dx = x sech-1 x- sin-1 x + C xcoth()= a? csch-1 x dx = x csch-1 x+ sinh-1 x + C ?ax γ sech-1()= 22a axa?x R b csch-1(x/a)=

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) x2x3xne=1+x+++…++ … 2!3!n!x?axa2?x2 β α 餘弦定理: a2=b2+c2-2bc cosα b2=a2+c2-2ac cosβ c2=a2+b2-2ab cosγ c 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??1= n i?1nn(?1)nx2n?1x3x5x7sin x = x-+-+…++ … (2n?1)!3!5!7!(?1)nx2nx2x4x6cos x = 1-+-+…++ … (2n)!2!4!6!(?1)nxn?1x2x3x4ln (1+x) = x-+-+…++ … (n?1)!234(?1)nx2n?1x3x5x7tan x = x-+-+…++ … (2n?1)357-1?i= ?n (n+1) i?1n?i2= i?1n1 n (n+1)(2n+1) 6?ii?13= [?n (n+1)]2 Γ(x) = ?t0?x-1-te dt = 2?t0?2x-1?t2edt = ??01(ln)x-1 dt t(1+x)r =1+rx+?1r(r?1)2r(r?1)(r?2)3x+x+… -1

大寫 Α

??0xm?1dx (1?x)m?n小寫 讀音 alpha 大寫 Ι 小寫 讀音 iota 大寫 Ρ 小寫 讀音 rho α ι ρ

Β Γ Δ Ε Ζ Η Θ β γ δ ε ζ η θ beta gamma delta epsilon zeta eta theta Κ Λ Μ Ν Ξ Ο Π κ λ μ ν ξ ο π kappa lambda mu nu xi omicron pi Σ Τ Υ Φ Χ Ψ Ω σ, ? sigma tau τ υ upsilon phi φ khi χ psi ψ ω omega

倒數關係: sinθcscθ=1; tanθcotθ=1; cosθsecθ=1

sin?cos?; cotθ= cos?sin?平方關係: cos2θ+ sin2θ=1; tan2θ+ 1= sec2θ; 1+ cot2θ= csc2θ

商數關係: tanθ=

順位高; ? 順位高d 順位低 ;

順位低101?0*? = *? = = 0* = 00??00 = e0(??) ; ?0 = e0?? ; 1? = e0??

順位一: 對數; 反三角(反雙曲) 順位二: 多項函數; 冪函數 順位三: 指數; 三角(雙曲) 算術平均數(Arithmetic mean) 中位數(Median) 眾數(Mode) 幾何平均數(Geometric mean) 調和平均數(Harmonic mean) X1?X2?...?Xn n取排序後中間的那位數字 X?次數出現最多的數值 G?nX1?X2?...?Xn H?1 1111(??...?)nx1x2xni平均差(Average Deviatoin) 變異數(Variance) ?|X1n?X|n ?X)2?(X1nin or ?(X1ni?X)2n?1 標準差(Standard Deviation) ?(X1ni?X)2n

or ?(X1ni?X)2 n?1