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1、考无忧论坛考辅幣理版高等数学微分和积分数学公式（集锦）（精心总结）111DA-COn = mn in（系数不为0的情况）二、重要公式（1）hm曲丄=17 X(3) lim亦(d o) = l n-oo(4) lunn = 1n-oo(7) limaiccotx= 0A-(10) hm “ = s(5) limaictanx = 8 2(8) lun aiccotx = x-co(11) lim x1 = 1A-0*lim arc tan x =y2(9) hm ex =0三、下列常用等价无穷小关系（x - 0）tanx xarcsinx xaictan x x1-cosxxln(l + x)x
2、 ”-lxa” 一1 xlna(1 + x) _ 1 dx四、导数的四则运算法则(w v)r = U V9(wv) = liv+UVUV-lIJ V-五、基本导数公式(l)(c) = 0 x = “xT(3)( sill x) = cosx(4)(cosxj = - sin x(5)(taiix/ = sec2 x（co（.r j = -CSC2 X(7)( sec x) = sec x-tan x(8)(CSC a)=-cscx cotx9()=ex9(10)(/) =ax Ina(11)(111 x)=考无忧论坛考辅滋理版叫)=池(13) (aicsinxy =、yjl-x2(14) (a
3、iccosx)r =-li-x2(15) (aic tail x)=11+x2r/(16)( aic cot x) = (17)(兀)六、高阶导数的运算法则(1) w(x)v(x)(n)= /(x)fnv(x)(n)(2) M(x)F = cd)(x)(3) M(Q+b) = a忖川(ax+b) (4) ”a)2(x)r = f c仙心)(x)网(兀)A0七、基本初等函数的n阶导数公式(4丿sill (d.Y+/?)=a91 sin ax+h+n 2(5)cos(ox+b)丁=an cos ax + b+n 2(处+ b)“ln3 + b)叫(一1厂目畔(ax+b)八、微分公式与微分运算法则(
4、l)d(c) = 0 d(x) = .严 dx(4) J(cosx) = - sinxdx(5) J( tanx) = sec2 xdx(3)d (sin x) = cos.vvd(cotx) = -esc，xdx(7) J(secx) = sec x-tan xdx(8) JI esc x) = - esc x - cot xdx d(ex)=eKdx(10)d(a、)= ax Inadx(11) J (In x) = dx X(12)J(log/) =dx xlna(13) d (arc sin x) = =dxJ-x2(14) d (arccosx) = -dx(15) J(aictan
5、x) =( d(aicco(x) = -九、微分运算法则(l)d (wv) = Jw+Jv d(cu) = cdu考无忧论坛考新滋理版(3) d(uv) = vdu + udv十、基本积分公式(1) |Wv= kx+cf严 xdx =+c” + 1“帖侖+Csin x+c(7) jsinxtZv = -cos.r+c(9) = esc2 = -cotx+ c SIU X J(8) ；dx = f sec2 xdx = tanx+ cJ COS X J(10) f dx = aictau x + c1+r(11) f dx = aic sin x+c十一 下列常用凑微分公式积分型换尤公式jf(a
6、x+b)dx =丄打+b“(Q + b)u = ax+b”刊叫x冷”（讪鬥u = xj/(lnx)-dx= J/(liiA/(ln.v)lu = hixu 二 ex”的皿匸总咖）u = aj f sinx) cosxdx = j f (sinx)d(sinx)U = Sill Xj f (cos x)-sin xdx = -j/(cosx)rf(cosx)M =COSXj f 1 tanx)sec2 xdx = j/( tanx)d(tailx)u = tail xj/(cotx)-csc2 xdx = j/(cot.r)J(cotx)u = cot Xj f 1 arctan x)dx =
7、j /(arcft7nx)rf (arc ta n x)u = aictan xf f (aicsinx)- . 1 Jx= / (aicsin x)J (aicsin x)yjl-x2Ju = arcsinx考无忧论坛绑整理版I二、补充下面几个积分公式| tan xdx = - In |cos x + c| sec az/a = ln|sec tail x +cf亠=丄az沁+c+ jt a asin x + cJcsc.S = In esc x-cor x + cz dx = arcsin+cJ 4a1 - x2a十三.分部积分法公式(1)形如疋0“.丫，令“ = x”，dv = edx
8、形如 Jxn sin xdx 令 ” = x, Jv = suixr形如 J xn cosxdx 令 u = xndv = cosxdx形如 Jx aictanjjj令=aictan xdv = xndxxn lii xdx 令 “ =hi x,dv = xndx形如 J严 sinxdx, J严cosxdx 令二严,sinx,cosx 均可。十四、第二换元积分法中的三角换元公式 J/ -x2 x =a sin/yjcr +F x = atant(3)JF - a2 x= a sec t【特殊角的三角函数值】(1)sinO = 0(2)7t Sill =丄亠更n .(4) Sill = 1 )(
9、5) sin= 062322(1)cosO = 1(2)n cos=巫n 1(3) cos =-(4) cos= 0 )(5) cos = -l623 22(1)tan 0 = 0(2)tan =_x/3(3)tn li = -3(4) tan 不存在(5) tan /r = 06332(1)cotO不存在(2)cot =V3(3) cot = - (4) cot = 0(5) cot 不存6332在考无忧论坛考辎幣理版I五、三角函数公式 1两角和公式sin(A- B) = sin AcosB-cosA sm Bsm(A + B) = sinAcosB + cos A sin Bcos(4 +
10、 B) = cos4cossin Asin Bcos(A- B) = cos 4 cos + sin A sin Btan(A 一 B)=tan A 一 tan B1 + tau A tau Btau(A + B)=tan A + tan B1 一 tan A tail Bcot(4 + B)=cot A-cot -1cot +cot 4cot(A-B)=COXAB+1cot 5-cot 42二倍角公式sjn2A = 2sjb Acos4cos2A = cos2 A- sin2 A = 1- 2sin2 A = 2cos2 力一 1tan 2A =2 tan A1-tan2 A3半角公式.A
11、/1-cosAsm = J2*2Acos=2fl + cosAV 2A /l-cos4 tau = J2 V 1 + cos Asin A1 + cos AA1 + cosAcot = J2 v 1-cos Asin A1-cos A4和差化积公式. c a + ba-bsin a + sin b=2 sincos2 2. 宀 a + ba-bcosa + cosp= 2coscos2 2. c a + b . a-bsina-sinb= 2cossin2 2f . a + b . a-bcos a cos b = -2 sin sin2 2tan a + tan h =sin (a + b)
12、cosa cos/?cosacosb =5积化和差公式sin asinb = 一cos( a + b)-cos(a-b) siu(7C0s/? = sin(a + /?)+ sm(a-b)1 ricosacosb = cos( a + b) + cos(a-b) cosrjsin/? = sin(a + /?)-sin (a b)考无忧论坛考辅滋理版6万能公式sin a =r a2(an 21 + tail 2cosa =1+tair-2tan a =r a2 tan 2= a1 _ fair 27平方关系sill2 x+cos2 x= 1sec2 x-tan1 x= 1esc2 x-cot2 x= 18倒数关系 tanxcotx = 19.商数关系sin xtan x =cosxsecxcosx = 1cscx sn】x = 1cotxcosx sin xI 六、几种常见的微分方程z/y1 可分离变童的微分方程：=f(x)g(y) . Z(x)gl(y)dxf2(x)g2 (y)Jy = 02.齐次微分方程：字“住、ax xj3阶线性非齐次微分方程：-+p(x)y = Q(x)解为：