微積分復習資料——公式

1、微彩今(上號習資料公式1函數(shù)初等函數(shù)：常量函數(shù)y二C(C)幕函數(shù)y=xa(a)指數(shù)函數(shù)y=ax(a0, a=#0)對數(shù)函數(shù)logax(a0, aO)三角函數(shù)y二sinx y = cos x y = tan x y = cot x反二角函數(shù)y二arcsi n x二s in_ 1y = arctan x = tan x y = arccot x = cot三角函數(shù)公式1兩角和公式siii(A + 3) = sin A cos B + cos A sm Bsin(A一B) = sin A cos B - cos A sin Bcos(4 + B) = cos A cos B - sin A sin

2、Bcos(A -B) = cos A cos B + sm A sm B2.二倍角公式cos2A = cos2A-sm24 = 1一2sin A = 2cos2A-lA ll + cosAcos = J-2 V 2A /1-cos Asin Atan =-=-2 V1 + cos A 1 + cos AA11 +cos Asin A2 Yl-cosA 1-COSAtan(A + B) =tan A + ianB1 - tail A tail Btan A - tan1+tan A tail Bcot B + cot 4cot(A _ B)=cot A cot B +1cot - cot Ay二

3、arccos x二cossm 24 = 2 sm A cos Atan 24 =2 tail A1-tair A3.半角公式5.積化和差公式sin c/ sin b =-丄cos (a + b) - cos (a-b)sinacosb = *sin(a + b) + sm(a-b)6.萬能公式cos a cos b =丄cos (a + b) + cos (a - b)1r ncos a sin b = - L sin (a + b) -sui (d - b)【特殊角的三角函數(shù)值】X0Tt*6Tt3Tt2Tls in x012210cos X1A/3T120-1tan x0羽3麗不存在0cot

4、 x不存在30不存在4.和差化積公式 .f c . a+b a-bsmf/ + snip = 2sm cos2 2sm-sm/? = 2cosci + bFsuia-bFcosd + cosb = 2cosa + bI-cosa-bI-cos ci - cos b = -2 sma + bI- sma_bFtan + tan/?=sm(d + b)cos。cosb2 tan 2sm ci =- -1+tan227.平方關系sm2x + cos2x = 18.倒數(shù)關系taiix-cotx = l9.商數(shù)關系sinxtanx =-cosxcosa =.2a1 tail 21 + tail222 t

5、ail 9 tail a =-1-tan22sec2x-tan2x = lCSC2x-cot2x=secx cosx = lc5cx-smx = lcosxcot x =-smx2極限數(shù)列極限四則運算 若數(shù)列& 與 為收斂數(shù)列, 則an bn an?bn也是收斂數(shù)列，且I im (a門 士bj = I im士I im bnnTcop conT coI i m (an?bn) = I i m an? I i m bnp oop co叫第bn羊0及I im b門I im b” ()n-*函數(shù)極限運算 定理1四則運算法則I i m f (x) g(x) = I imf(x)士I i m g(x) =

6、 A BC-*XOXXoXTxo定理2復合函數(shù)極限 設函數(shù)y = f 4)(x)是函數(shù)u = 4)(x), y = f(u)的復合函數(shù)。I im 4)(x)二uoI imf(u)二uo若ft), y = f (u)在UO有定義且uTuo,則I imf g(x) = f (UQ)I im (b (x)二un因為LX。，所以定理結(jié)論也也可寫成I imf 4)(x) = f I im 4)(x) X-*xoxTx。I imf(x)推論3若XTXO存在，C為常數(shù)，則I imCf(x) = Cl imf(x)X XoXTxoI imf(x)推論4若xTxo存在，n為正整數(shù)，則I imI im f(x)?

7、g(x) = I imf (x)? I im g(x) = A?BxTxxTxoI irr f (x)I irr g(x)g(B * 0)CO常用XTO時的等價無窮小sinxx, arcs in xx, t an xxFarc tanxln(1 + x) x, ex- 1x, 1 - cos x冷,ax- Cxln a, (1 + x)a-廣ax常用極限sin x1 XH m (1 + -)二eXT8limy/a(ci o) = lliin yfn = 1lunaictanx = V-X2lim arcianx =XT-R2Iimf(x)n二xTxI imf(x)nXXQlimarccotx

8、= 0AT*liin aic cot x=兀A-xlim ev= 0.VTYCliin ex= ooA-+Xlull xY= 1A-0*b0係數(shù)不為0的情況）3導數(shù)導數(shù)的四則運算法則 (U V) = ur V*(uv) = U V + uvr(Cu)二Cl/,推廣(uvw) = u VW + UV1W + UVW反函數(shù)導數(shù)：f(x)=希7或?qū)?1忌 復合函數(shù)導數(shù)：y(x) = F(u) + b(x)或月二務囂(鏈式法則)基本導數(shù)公式 =0 x= “.嚴(3)(sinx) = cosx(4)(cosx) = - sin x(5)(tanx) = sec2x(6)(cotx) =-csc2x(7)

9、(secx) =sec x-taiix(8)(cscx) =-cscxcotx (d JFind(Q)ex) =ex(13)(aicsinx)(14)(aiccosx)(15)( arctail x)=(16)(aiccotx)1ll-X211 + x211 + x2(17)(%/ =1高階導數(shù)的運算法則(1)/(X)V(X)(，)=W(X)HV(X)(，)(2)cu(x)F)= cd)(x)(3)“(ax+/?) = ad)(ax+b)p心).咻)=f制宀代i=0基本初等函數(shù)的n階導數(shù)公式 a”）（）=/(3) (ajJd=cinsin ax + b + n蘭I 2丿(5) cos + b)

10、y)= a cos ax + b + n- z(1 Y；(1Vr宀！命=(_1)(zr1+呵=(_1廠笄褂(ax+b)5微分微分的四則運算根據(jù)與導數(shù)的關系，所以與導數(shù)相同微分的近似計算中的應用由函數(shù)增量與微分的關系？y二f(xo)?x + a?x二dy + a ?x,其中？xTO時a TO,當|?x|很小時，有?y 4 dy,因此f(x + x0) = f(xo) + f (xo)?x或當x a xo時有f(x) a f(x0) + f (xo)(x - x0)令x0 = 0,得下列函數(shù)在原點附近的近似公式：sinx a tanx = Inxx, ex1 + x微分公式與微分運算法則(1)J(

11、c) = 0心)=代加(3) d (siii x) = cos xdx(4) d (cos x) = - siii xdx d (tail x) = sec2xdx(cot x) =一esc2xdx(刀(7 (sec x) = sec x-tail xdx(8) d (esc x) =一esc x cot xdx d () = edx微分運算法則 d (u u) = du土du d (wv) = vdu + udv幾種常見的微分方程（課外知識）(10) d(d ) = avIn MY(11) J(lnx) = x2)6/(log/) = -l-(13) cl (arcsui x)=丄_dxy

12、1 X(14) d (aiccos x)=/ dxyjl-x2(15) d (aictan x) = dx1+對(16)- d (arc cotx) =- dx1 +(4)dgz 丿vdu一udvV1可分離變量的微分方程： =/a）g（y） ,/（X）（y）&+厶$（），妙=02.齊次微分方程：字上、ax x丿3. 階線性非齊次微分方程：字+P（x），= 0（x）dx解為：y =eP（x）dxQ（x）ePdxdx+c8不定積分基本積分公式2、(X(,dx = + cJa + l14、J tail =-lii cosx| + c3、jtr = ln|.v| + cX15、J cotAZZT= l

13、ii|siii x + c16、J sec xdx = lii|sec x + tan x + c4、5、exdx = ex+ ccscxdx = n cscx-cotx + c = lii tan +c26、sinxdx = -cosx+c7、Jcosxzir=sinx + c18、_-dx=丄arc tan + cJ cr+JTa a8、9、f dx= sec2xdx = tanx + cJCOS XJfdx= fcsr xdx = -cotx+cJ sui x19、20、10、J secx tan xdx = secx+c11vcscxcoixdx = -cscx+c2K12、f= ar

14、csuix+c22、23、J shxdx = chx + c其中shx =-24、1 i 1 . a + x - -dx = lii-a -x 2a a-x_ dx= 111 x一CT 2ax-ax+ci+ c課外fdx=arcs in +cyla2- XTa1lx2crdx= In x+yjx2a2+c為雙曲正弦函數(shù) （課外知識）chxdx=血+c其中c/u =今二為雙曲余弦函數(shù)（課外知識）乙F列常用湊微分公式積分型換元公式j f(ax+b)dx= f (ax+h)d (ax+h)u = ax+bj xf(ax2+ b)dx = f (ax2+ b)d(ax2+ b)u二ax + bj*(&

15、)dx二2J(jx)d(&)u二&J*-f (1 n x)dx = J*f (1 n x)d(l n x)u = 1 n x“(*=* jv(# w(*)課外11 = pJ/*(illx)丄t/x = |/(llix)rf(111x)課外Xu = nxfexexdx = fexyiex)u = exJ7)心沽課外u = axj/(siiix)-cos xdx = |/(siii (sill x)w = siiixJ f (cos x) sin xdx = -j/( cos xl (cos x)u = cos Xj/( tan x) sec2xdx = j/( tan x)d (tan x)課外w = tanxJ / (cot x) esc2xdx = j/(cotx)/(cotx)課外w = cotxj f (aictaii x) dx = /(arc ta nx)t/( aic tanx)1 i Xw = aictanxj f (arcsin x) /】dx = f( arcsui x)d (arc sin x)y/l X2vu = arcsin x分部積分法公式形如j xneaxdx,令u = xn,