1) C = 0 ⑵ 0a a—1 ( ) ( )= axa —1 (3) a x = a x ln a (4) e x⑸bg ax =盂(6) Gn x) = 1 (7) (sin x) = cos x (8) (cos x) = — sin xx(9) (tan x) = sec2 x (10) (cot x) = — csc2 x (11) (sec x) = sec x - tan x(12) (cscx)=-cscx-cotx (13) (arcsin) = -1— (14) (arccosx) = — -1—U-x 2 %'1-x 2(15) (arctan x) = 1 (16) (arc cot x) = — 11+ x2 1+ x2(),= U 'V — U 'V V V2土 v)= U ±『 (u - v)= U - v + u - V{f tp(x M = f，tp(x )].0(X )(1) J 0dx = c (2) J xa dx ==1a+1=ln xl + cxa+1 + c("—】)⑶ J axdx = + ax + c(a = —1) ln a(4) J sin xdx = — cos x + c (5) J cos xdx = sin x + c (6) J tan xdx = — ln cos xl + c(7) J cot xdx = lnlsin x + c (8) J sec xdx = lnlsec x + tan xl + c (9) J csc xdx = lnlcsc x — cot xl + c(10) J sin2 xdx = + x 一 丁 sin2x + c (11) J cos2 xdx =青 x + 丁 sin2x + c(12) J sec2 xdx = tan x + c(13) Jcsc2xdx = — cotx + c (14) JT dx = arctanx + c (15) J 1 dx =十arctan^ + c1+x2 x2 +a2 a a(16) J~1- dx = +lnx2—a2 2ax—ax+a+c (17) I-=^= dx = arcsin x + c (18) J 1 dx = arcsin x + c 1—x2 '；a2 — x2 ac (2) J^^=、'x2 - a 2(19) J ,— dx = ln x + * x2 + a2 +Vx 2 + a 2(3)(4)(1)(2)(6)(7)(8)(9)(10)=-arccos x + c(11)对这些公式应正确熟记.可根据它们的特点分类来记.公式（i）为常量函数o的积分，等于积分常数c.公式（ 2）、（3）为幕函数，一 的积分,应分为u与时，积分后的函数仍是幕函数，而且幕次升高一次.特别当时，有CE = -1时，公式（4）、（5）为指数函数的积分，积分后仍是指数函数，因为3 _ "）式右边的不在分子，应记清.是一个较特殊的函数，其导数与积分均不变.应注意区分幂函数与指数函数的形式，幂函数是底为变量，幂为常数；指数函数是底为常数，幂为变 量.要加以区别，不要混淆.它们的不定积分所采用的公式不同.公式 式.6）、（7）、（8）、（9）为关于三角函数的积分，通过后面的学习还会增加其他三角函数公公式10）是一个关于无理函数的积分11）是一个关于有理函数的积分公式=J -—dx = arctgx + c = -arcctgx + c[. a xn + a xn-1 + + a、lim 1 n = vxTg b xm + b xm-1 + + b0 1 maob00系数不为0 的情况）sin x二、重要公式(1) lim = 1xT0(4) lim o) = 1nTg兀(6) lim arc tan x =-—2xT-89) lim ex = 0xT-g7) lim arc cot x = 0xTg(8) lim arc cot x = nx T-g10 ) lim e x = gxT+811) lim xx = 1xT0+三、下列常用等价无穷小关系sin x □ xtan x □ xarcs ixri] x1arctan x □ x 1 - c o x □ x 22ln (1 + x ) □ xex -1 □ xax -1 □ x In a(1 + x》-1 □ dx四、导数的四则运算法则(u 土 v ) = u '土 v '(uv ) = u 'v + uvu v- uvv2五、基本导数公式(1) (c ) = 0(2) X卩=卩X卩-1⑶(sin x) = cos x(4) (cos x) = - sin x⑸(tan x) = sec2 x⑹ (cot x) = - csc2 x(7) (sec x) = sec x - tan x⑻(csc x) = - csc x - cot x⑼⑽二 ax In a(11) (ln x)'=-x(12)x)= aX ln a(13) (arcsin x )=(15) (arctan x )'= -—1 + x 2(16) (arc cot x )' = - 一1—(⑺1 + x 21 - X 2六、高阶导数的运算法则1)u (x )± v (x )](n) = u (x Xn) + v (x )(n)2)cu(n) (x)(3) u (ax + b )(n)== a nu (n)(ax + b )⑷ u (x )• v (x )](n )=2 cku (n-k )(x [ (k) (x)nk=0七、基本初等函数的n阶导数公式(1 ) Cn)n)= n ! (2) Cox+b )n) = an • e«x+b(3)(ax)(n) =axlnna(4) sinsin (ax + b )](n) = an sin ax + b + n •—(5) cos(ax + b)](n) = an cos ax + b + n •—2丿(6)(1 \(n) = ( 1) an • n!(ax + b)n+1(7)[ln (ax + b ) (n) = (-1)(n - 1)!»n-1 罕_ (ax + b )n八、微分公式与微分运算法则1 d (c ) = 0 ⑵ d (x J =卩 x p-1dx(3) d (sin x ) = cos xdx4 d(cosx)=-sinxdx (5) d (tan x) = sec2 xdx⑹ d (cot x) = -csc2 xdx7 d (sec x) = sec x • tan xdx(8) d (csc x )= - csc x • cot xdx(9) d (ex )= exdx⑽ d Cx )= ax ln adx1(11) d (ln x )= — dxx(12)dGog x)= 1— dx (13)d(arcsinx)= 1——dx (14)d(arccosx)= — dxa x ln a J1- x 2 J1 — x 2(15) d (arctan x )= 1— dx1 + x 2(16) d (arc cot x ) = — —1— dx1 + x 2九、微分运算法则⑴ d （u 土 v ）= du 土 dv⑵ d (cu )= cdu⑶ d (uv) = vdu + udv十、基本积分公式⑴ J kdx = kx + c⑷ J axdx =竺 + cln a(5) J exdx = ex + c=vdu - udvv2J dx = ln |x| + cx(6) J cos xdx = sin x + c(7) J sin xdx = - cos x + c⑼ J丄= Jcsc2 xdx = -cot x + c sin2 x(8) J —1 dx = J sec2 xdx = tan x + c cos2 x⑽ J —1— dx = arctan x + c1 + x 2(11) J ； 1 dx = arcsin x + c1 — x2一、下列常用凑微分公式积分型换元公式J f (ax + b )dx = — J f (ax + b )d (ax + b ) au = ax + bJ f (x 卩)r—dx = — J f (x 卩》(x 卩)u = x卩J f (ln x )•1 dx = J f (ln x )d (ln x)xu = ln xJ f (ex )• exdx = J f (ex》(ex )u = exJ f Cx )• axdx =丄 J f Cx》Cx ) ln au = axJ f (sin x )• cos xdx = J f (sin x )d (sin x)u = sin xJ f (cos x )• sin xdx = -J f (cos x 为(cos x)u = cos xJ f (tan x)• sec2 xdx = J f (tan x》(tan x)u = tan xJ f (cot x )• csc2 xdx = J f (cot x》(cot x)u = cot xJ f (arctan x)・ 1 dx = J f (arc ta n x》(arc ta n x)1 + x 2u = arctan xJ f (arcsin x )• 1 dx = J f (arcsin x》(arcsin x)弋 1 - x 2u = arcsin xf tan xdx 二-ln|cos x| + cJ sec xdx 二 ln |sec x + tan x| + cf cot xdx = ln |sin x + c f csc xdx 二 ln |csc x - cot x| + c1 1 xdx = arctan + ca 2 + x 2 a aJ 1x2 - a 2dx = 1 ln2adx =xarcsi。