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1、一.基本初等函数求导公式(cy = o(2)(six)r =cosx(4)(cosx) =-sinx(tanx)r = sec2 x(6)(cot x)f = -esc2X(secx)r = sec x tan .v(8)(cscx)f = -cscxcotx(ax y = ax In a(10)(ery = ev(11)(log”)-xm a(12)(In x)f =(firpcin y9 (firr*r*oc vV 1(13)vl -X2(14)v Wo 人 fJl -X(15)(arctan x)r -1、1+JC(16)(arc cot x =-11 + x2函数的和、差.积.商的求导
2、法则设“ =/心),V = v(x)都可导,则()(M V) = ll v(2)Cuy = cu(C 是常数)(3)(uv)f = ufv + uvf(u 1fufv-uvr(4)”2反函数求导法则若函数*=0(刃在某区间厶内可导、单调且0)h ,则它的反函数)=f(X)在对应区间人内也可导,且dy 1 dx dxdy复合函数求导法则设= /(),而“=卩(尤)且/()及#(x)都可导,则复合函数y = /卩(0的导数为dy _ dy dudx du dx 或 yf =广()0(x)二.基本积分表(1) kdx = kx + C(k 是常数)(2)(心1)(3) -dx = nx+CJ x(4
3、) 人=arl tanx + C1 + JTdx= arcsinx + C(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)j cos xdx = sin x + C lx = tan x + C cos* x= -cot x + C siir xJ sec x tan xdx = sec x + C| esc x cot xdx = - esc x + Cexdx = ex +CC , (a 0,且a H 1)nashxdx = chx + Cj chxdx = shx + Cf dx = arc tan + C 犷+对a
4、f - dx = In IJ x2-a2la口 l+C2ax + a.(lx = arc sin + C-Ja2af . J dx = ln(x + y/a2 + X2) + CJ yja2 +x2dxJ tan xdx = -In I cos x I+C(22)J cot xdx = In 1 sin x 1+C(23)J sec xdx = In 1 secx + tan x 1+C(24)f esc xdx = In 1 esc x - cot x 1+C注:1、从导数基本公式可得前13个积分公式,(16)-(24)式后儿节证。2、以上公式把天换成仍成立,是以x为自变量的函数。3、复习三
5、角函数公式: 22(夕(J.rr.r1 + COS2sin x + cos x = 1, tan x +1 = sec x. sin 2x = 2sin xcos x. cos* x =21-cos 2x2注:由J f(pM(px)dx = f(px)d(p(x),此步为凑微分过程,所以第一 类换元法也叫凑微分法。此方法是非常重要的一种积分法,要运用自如, 务必熟记基本积分表,并掌握常见的凑微分形式及“凑”的技巧。小结:1常用凑微分公式积分类型换元公式1.J /(ax + b)dx =丄 j/(ux + b)d(ax + b) (“ H 0)u = ax+ b2 J f(x/i)xpdx =
6、-(“ H 0)u = x3.J* /(In a)-丄x = J/ (In x)J(lnx)u = In x第4.J/(ev) - exdx = jf(ex )dexu = ex换5ddx =1* f(ax)daxu = ax元6.J/(sin a) cos xdx = J/(sin x)d sin xit = sinx积 分7 J /(cos x) sin xdx = - J f (cos x)d cos xu = cos X法8. J f (tan x) sec2 xdx = J/(tanx)Jtan xu = tanx9.J /(cot x)csc2 xdx =/(cotx)J cotxu = cot X10.j f (arct an x) *- dx = J/(arct anx)J(arct ail a)u = arct an x1 l.J f (arcs in a)!dx = 一 J /(arcs in a) J (arcs in a)u = arcs in x