商业微积分公式

1、A thesis submitted toin partial fulfillmentof the requirementfor the degree ofMaster of Engineering有关高等数学计算过程中所涉及到的数学公式(集锦)im +4严 + + © x bxm + bvxm + + %=< 0（系数不为0的情况）二、重要公式呼字“(2 ) lim(l + x)7=e(3)iny/a(a > o) = ” f X(4 ) lim /n = 1“TOO.兀lim an?kmx = - 2(7) lim arc cot x = 0(5)lim arctan

2、 x = 2(6)(10) lim ex =qo(8) lim arccotx = (11) lim xx = 1xt(t(9) lim ex = 0XT-X三、下列常用等价无穷小关系（xtO）arctan x x1-cosx xsin x xtan x xarcsin x xln(l + x)兀ax 一1 xna(l + x)。_1 &四、导数的四则运算法则(mv) =ufv + uvfx"=时'(5)(tanx) = sec2 x(w±v) = U ± V五、基本导数公式(c)'=0(4)(cosx) =-sinx(3)(sinx) =

3、cosx(6)(cot x) =-csc2 x(7)(secx) =secx-tanx(8)(cscx) =-cscx cotx(10)(/j=aTna(lD(lnxj= -x(13)( arcsin x)' = /、a/1-a-2(14)(arccosx)' = - pJ_yj-x2(15) (arctan x j =-,(16) (arc cot = -】、(x) = 1 (18)六、高阶导数的运算法则(1 ) «(X)±V(X)(，)= w(x)l， ±V(xy/0(2)=c“""(x)(3) "(ov+b) &

4、#39;(l =a"u,! (tix + b)(4 )川=工(x)A-0七、基本初等函数的n阶导数公式 X)何=巾!(2)(严)严/cos (心+ b)(小(4) sin(ax + /?) “ = a11 sin ca + b + n- =a11 cos ax+ b + n |I 2丿In(俶+ b)=(l厂汀畔ax + h八、微分公式与微分运算法则(l)d(c) = O(2)(“)= “十加(3) d (sin x) = cos xclx(4)(cosx) = -sinx厶(5)d(tanx) = sec，xdx (6) J (cot x) = -csc2 xdx(7)d(secx

5、) = secx tanxr(8) d (esc x) = esc x cot xdx dex) = exdx(10)(/)FIn adx(ll)J(lnx) = ljxX(og')=dx(13)d (arcsin x) =； dxcl (arccos x) = - , = dx x/l-T?(15)d (arctan a) = -，-dx(16) d (arc cot x ) = - - < dx九、微分运算法则(1) du±v) = du±dv(2) (cu) = cdit d (wu) = vdu 4- adv(4)d/ u、vdu _ uclv<

6、vjV2十、基本积分公式(2) J xpdx =严+ c"+1(3) j = In |x| + cX(4) J % =+ c(5) Jexdx = ex + c(6) J cos xclx = sin x + c JsimzZv = -cosx + c(9) fr- = fcsc2 xdx = -cotx + c(siirx(8) f 一dx= f sec2 xdx = tun x + c J cos' XJ10) J 1 , dx = arctan x + c(11) J j 1 dx = arcsin x + c十一、下列常用凑微分公式积分型换元公式J f (ax + b

7、)clx = j f (or + /"/ (ov + /?)w =ax+b加冷”（兀字3“）u = x"j/(lnx)-6/x = j/(lnx/(lnx)X“ = lnxu = exu =axJ /(sin x) cos xdx = |/ (sin xl (sin x)u =sinxJ f (cos x) sin xdx = - J /(cos x*/ (cos x)U = COS Xj/(tanx) sec2 xdx = J/(tanx)t/(tanx)u = tan xJ f (cotx)esc2 xdx = |/(cotx)c/(cotx)u = cot Xj /&

8、#39; (arctan x) (仪=J / (arcta nx)t/( arc ta n a )u = arctan xf f (arcsin x) - ,dx = f / (arcsin a (arcsin x)u = arcsin x十二、补充下面几个积分公式J cot xdx = In |sin x + c| esc xdx = ln|csc x-cot x| + cJ tan xdx = -ln |cos + cJ sec xdx = ln|sec x + tan x| + cp 1.1x-dx = arctan + cdx = urcsin + caa=dx = In x + Jx

9、2 ±a2 + c±7十三、分部积分法公式 形如 JV严dx,令 h = x dv = eaxdx形如 Jsin xdx 令 ii = xn , dv = sin xdx形女口 J xn cos xdx 令 u = xn , dv = cos xdx形如 Jx11 arctan xdx ,令 u = arctan x 9 dv = xndx形女叮 xn In xdx,令 “ =In x, dv = xndx形如 Jeax sin xdx Jeax cos xdx 令"=eClX ,sinx?cosx 均可 o十四、第二换元积分法中的三角换元公式(1) Jc, -F

10、x = asint(2) cr +x2x = at:mt(3) Jx1x = asect【特殊角的三角函数值】(1) sin 0 = 0(2)7t sin =612(3) sin- = (4)sin = 1 )2(5)32sin/r = 0(1) cos0 = l(2)7tCOS =羽(3) cos ：_ 1(4)cos = 0 ；(5)623 22cos/r = 1(1) tan 0 = 0(2)兀 tnn =6筋3(3 ) tan =3J(4)tan 不存在2(5)tan /r = 0(1) cot 0不存在(2) cot :(3) cot _V3(4)cot = 0 ( 5 )cot/r

11、6332不存在十五、三角函数公式1两角和公式sin(A + B) = sin A cos B + cos A sin Bsin( A - 3) = sin A cos B 一 cos A sin Bcos( A + B) = cos A cos B - sin A sin Bcos(A - B) = cos A cos B + sinA sin Btan(A + B)=tan A + tan B1 一 tan A tan Btan(A-B)=tan A 一 tan B1 + tan A tan Bcot(A + B)=cot A cot 3 1cot B + cotAcot(A-B) = CO

12、McOtB + 1cot cot A2.二倍角公式sin 2A = 2sin AcosAcos 2A = cos2 A-sin2 A = l-2sin2 A = 2cos2 A-ltan 2A =2 tan A1-tan2 A3. 半角公式.A /1-cosAsin 7 _ V 2A /1-cos A sin Atcin = J=2V I + cos A 1+cos AA /I + cos A sin Acot =2 v 1-cos A 1-cosA4. 和差化积公式sin« + sin/? = 2sincosa-hcos a + cosh = 2coscosa_b"T&q

13、uot;., c a+b . a_bsin a-sin b = 2 cossin2 2f c . a + b . a-bcos a 一 cos Z? = -2 sin sin2 2tan a + tan /?=sin(a + Z?)cos " cos/?5. 积化和差公式sin°sinZ? = 一* cos(d+b)-cos(a-b)1 r n sinocosZ? = Lsin(a + /?) + sin(a-b)1 rcosi/cos/? = cos(a + ) + cos(a-Z?) cosasinb = sin(a+Z?)-sin(d-Z?)cos a =2 tan 2 tan a =1-tan2 26. 万能公式2 tan ? sin a =1 + tan2 27. 平方关系sin2 x + cos2 x =