微积分公式大全

微积分公式

Dx sin x=cos x ? sin x dx = -cos x + C sin(-x) = -sin x -1-1 cos x = -sin x ? cos x dx = sin x + C cos(-x) = ? - cos x 2-1-1 tan x = sec x ? tan x dx = ln |sec x | + C tan(-x) = -tan x 2-1-1 cot x = -csc x ? cot x dx = ln |sin x | + C cot(-x) = ? - cot x sec x = sec x tan x ? sec x dx = ln |sec x + tan x | sec-1(-x) = ? - sec-1 x csc x = -csc x cot + C csc-1(-x) = - csc-1 x x ? csc x dx = ln |csc x – cot x | + C Dx sin ()= x cos ()= a-1-1-1-1xa?1a2?x2? sin-1 x dx = x sin-1 x+1?x2+C sinh-1 (x)= ln (x+a2?x2) -1-1? cos x dx = x cos x-1?x+C x?R ? tan-1 x dx = x tan-1 x-?ln (1+x2)+C xcosh-1 ()=ln (x+x2?a2) x-1-12ax?a? cot x dx = x cot x+?ln (1+x)+C tan-1 ()=2 aa?x2≧1 -1-1? sec x dx = x sec x- ln xx1a?x-1-1cot ()= tanh ()=ln () |x| <1 aa2aa?x|x+x2?1|+C sec-1 x1x?a? csc-1 x dx = x csc-1 x+ ln coth-1 ()=ln () |x| >1 |x+x?1|+C 22ax?a()= 22axx?aa2ax?ax?11?x2sech()=ln(+)0≦x2axx-1csc-1 (x/a)= ≦1 x11?x2csch ()=ln(+) axx2-1|x| >0 Dx sinh x = cosh x ? sinh x dx = cosh x + C duv = udv + vdu cosh x = sinh x ? cosh x dx = sinh x + C ? duv = uv = ? udv + ? vdu tanh x = sech2 x ? tanh x dx = ln | cosh x |+ C →? udv = uv - ? vdu coth x = -csch2 x ? coth x dx = ln | sinh x | + C cos2θ-sin2θ=cos2θ sech x = -sech x ? sech x dx = -2tan-1 (e-x) + C cos2θ+ sin2θ=1 1?e?xtanh x cosh2θ-sinh2θ=1 ? csch x dx = 2 ln || + C ?2x csch x = -csch x cosh2θ+sinh2θ=cosh2θ 1?ecoth x xDx sinh()= a1-13sin 3θ=3sinθ-4sinθ -1-12? sinh x dx = x sinh x-1?x+ cos3θ=4cos3θ-3cosθ 3 C →sinθ= ? (3sinθ-sin3θ) 322a?x→cosθ=?(3cosθ+cos3θ) -1-12? cosh x dx = x cosh x-x?1+ ejx?e?jxx-1cosh()= sin x = cos x = 2jaC -1-1jx?jx1? tanh x dx = x tanh x+ ? ln | e?e 22x?a1-x2|+ C 2? coth-1 x dx = x coth-1 x- ? ln | x?aex?e?x-1tanh()= 2 sinh x = cosh x = aa?x21-x2|+ C 2-1-1? sech-1 x dx = x sech x- sin x ex?e?xx-1β coth()= a+ C 2-1-1-1bca? csch x dx = x csch x+ sinh x x-1正弦定理:= ==2R sech()= sin?sin?sin?a+ C 222?a 余弦定理: a=b+c-2bc cosα γ 222a xa2?x2 b=a+c-2ac cosβ R b α csch-1(x/a)=c2=a2+b2-2ab cosγ c ?a xa2?x2sin (α±β)=sin α cos β ± cos α sin β sin α + sin β = 2 sin ?(α+β) cos ?cos (α±β)=cos α cos β ?sin α sin β (α-β) 2 sin α cos β = sin (α+β) + sin (α-β) sin α - sin β = 2 cos ?(α+β) sin ?2 cos α sin β = sin (α+β) - sin (α-β) (α-β) 2 cos α cos β = cos (α-β) + cos (α+β) cos α + cos β = 2 cos ?(α+β) cos ?2 sin α sin β = cos (α-β) - cos (α+β) (α-β) cos α - cos β = -2 sin ?(α+β) sin ?(α-β) tan??tan?tan (α±β)=, cot (α±?tan?tan??cot?cot?β)= cot??cot?x2x3xne=1+x+++…++ … 2!3!n!x?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?i= ?n (n+1) i?112i= n (n+1)(2n+1) ?6i?1n?ii?1n3= [?n (n+1)]2 (?1)nx2n?1x3x5x7tan x = x-+-+…++ … (2n?1)357-1Γ(x) = ?0te dt = 2?0tdt 1?x-1-t?2x-1e?t2dt = ?(ln)x-1 0?1t(1+x)=1+rx+-1

?xm?1dx m?n(1?x)大写 小写 读音 大写 小写 读音 大写 小写 读音 Α alpha Ι iota Ρ rho α ι ρ Β beta Κ kappa Σ β κ σ, ? sigma Γ tau γ gamma Λ λ lambda Τ τ Δ delta Μ mu Υ δ μ υ upsilon Ε nu Φ phi ε epsilon Ν ν φ Ζ zeta Ξ xi Χ khi ζ ξ χ Η eta Ο psi η ο omicron Ψ ψ Θ theta Π pi Ω θ π ω omega 倒数关系: sinθcscθ=1; tanθcotθ=1; cosθsecθ=1 sin?cos?商数关系: tanθ= ; cotθ=

cos?sin?平方关系: cos2θ+ sin2θ=1; tan2θ+ 1= sec2θ; 1+ cot2θ= csc2θ

順位高; ? 顺位高d 顺位低 ;

順位低0*? = 1?10 *? = = 0* = ??0000 = e0(??) ; ?0 = e0?? ; 1? = e0?? 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) 算术平均数(Arithmetic mean) 中位数(Median) 众数(Mode) 几何平均数(Geometric mean) 调和平均数(Harmonic mean) 平均差(Average Deviatoin) 变异数(Variance) 取排序后中间的那位数字 次数出现最多的数值 ?(X1ni?X)2n or ?X)2?(X1ni?X)2n?1 ?X)2标准差(Standard Deviation) ?(X1nin or ?(X1nin?1