高数微积分公式 【开心一刻】第2期：x

Dx sin x=cos x cos x = -sin xtan x = sec2 x cot x = -csc2 xsec x = sec x tan x csc x = -csc x cot x微积分公式? 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 | + Csin-1(-x) = -sin-1 xcos-1(-x) = ? - cos-1 xtan-1(-x) = -tan-1 xcot-1(-x) = ? - cot-1 xsec-1(-x) = ? - sec-1 xcsc-1(-x) = - csc-1 xx? 1Dx sin-1 x? 1? sin-1 x dx = x sin-1 x+1 ? x 2 +Csinh-1 ( x)= ln (x+a 2 ? x 2 ) x? Raa 2 ? x 2cos-1 ( x )=? cos-1 x dx = x cos-1 x-1 ? x 2 +-1 ( x)=ln (x+x 2 ? a 2 ) x≧1a? tan-1x dx = x tan-1x-?ln (1+x2)+Catan-1 ( xa)=? aa 2 ? x 2? cot-1 x dx = x cot-1 x+?ln (1+x2)+C? sec-1 x dx = x sec-1 x- ln |x+x 2 ? 1 |+Ctanh-1 ( xa)= 12 aln ( a ? xa ? x) |x| 1sec-1 ( x )=? asech-1( x)=ln( ? 1 +1 ? x 2)0≦x≦1acsc-1 (x/a)=xx 2a -1 ( xax)=ln( 1xx 2+1 ? x 2x 2) |x| >0Dx sinh x = cosh xcosh x = sinh x tanh x = sech2 x? sinh x dx = cosh x + C? cosh x dx = sinh x + C? tanh x dx = ln | cosh x |+ Cduv = udv + vdu? duv = uv = ? udv + ? vdu→? udv = uv - ? x = -csch2 x? coth x dx = ln | sinh x | + Csech x = -sech x tanh x? sech x dx = -2tan-1 (e-x) + Ccos2θ -sin2θ =cos2θ cos2θ + sin2θ =1csch x = -csch x coth x? csch x dx = 2 ln |1 ? e ? x1 ? e ? 2 x| + θ -sinh2θ =1 cosh2θ +sinh2θ =cosh2θxDx sinh-1()=-1( x )=a1a 2 ? x 21x 2 ? a 2? sinh-1 x dx = x sinh-1 x-1 ? x 2 + C? cosh-1 x dx = x cosh-1 x-x 2 ? 1 + Csin 3θ =3sinθ -4sin3θ cos3θ =4cos3θ -3cosθ→sin3θ = ? (3sinθ -sin3θ )→cos3θ =?(3cosθ +cos3θ )tanh-1( x )=a? aa 2 ? x 2? tanh-1 x dx = x tanh-1 x+ ? ln | 1-x2|+ C? coth-1 x dx = x coth-1 x- ? ln | 1-x2|+ Csin x =e jxe ? jxcos x =2 je jx ? e ? -1( x )=a? sech-1 x dx = x sech-1 x- sin-1 x + C? csch-1 x dx = x csch-1 x+ sinh-1 x + Csinh x =e x ? e ? x =e x ? e ? -1( x)=? aγ正弦定理:a=b=c=2Raxa 2 ? x ?sin ?sin ?csch-1(x/a)=? axa 2 ? x 2α余弦定理: a2=b2+c2-2bc cosαβcb2=a2+c2-2ac cosβc2=a2+b2-2ab cosγsin (α±β)=sin α cos β ± cos α sin βcos (α±β)=cos α cos β? sin α sin β2 sin α cos β = sin (α+β) + sin (α-β)2 cos α sin β = sin (α+β) - sin (α-β)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)2 cos α cos β = cos (α-β) + cos (α+β) 2 sin α sin β = cos (α-β) - cos (α+β)tan (α±β)=tan ? ? tan ?? tan ? tan ?, cot (α±β)= ? cot ? cot ?cot ? ? cot ?ex=1+x+ x 22!+ x 33!+…+ x n + …n!?ni ?11 = nsin x = x- x 33!+ x 55!- x 77!+…+ ( ? 1) n x 2 n ?1 + …( 2 n ? 1)!?ni ?1i = ?n (n+1)cos x = 1- x 22!+ x 44!- x 66!+…+ ( ? 1) n x 2 n + …( 2 n )!?ni=1262i ?1n (n+1)(2n+1)ln (1+x) = x- x 22+ x 33- x 44+…+ ( ? 1) n x n ?1 + …( n ? 1)!?ni3i ?13= [?n (n+1)]2-1x 3x 5x 7( ? 1) n x 2 n ?1? x-1 -t? 2x-12?1x-1tanx = x-+-357+…++ …( 2 n ? 1)Γ (x) =? t0edt = 2 ? t0e ? tdt = ?0(ln)dtt(1+x)r =1+rx+ r ( r ? 1)x2+ r ( r ? 1)( r ? 2 )x3+… -1 #