微分积分公式大全微分基本公式

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 < nin > 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

2、关系（x -> 0）tanx xarcsinx xaictan x x1-cosxxln(l + x)x £”-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" = “x"t(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

3、)(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) (aiccosx)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)fn>±v(x)(n)(2) m(x)f = cd")(x)(3) m(q+b)"" = a忖川(ax+b) (4) ”a)2(

4、x)r = f c仙心)(x)网(兀)a«0七、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(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

5、(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(aictanx) =( d(aicco(x) = -九、微分运算法则(l)d (w±v) = 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+

6、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(ax+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 = a"j f sinx

7、) 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 = j /(arcft7nx)rf (arc ta n x)u = aictan xf f (aicsinx)- . 1 jx= / (aicsin x)j (aicsin x)yjl-x2ju = arcs

8、inx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 形如 jxn sin xdx 令 ” = x", jv = suixr形如 j xn cosxdx 令 u = xndv = cosxdx形如 jx" aictanjjj令

9、"=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 )(5) sin= 062322(1)coso = 1(2)n cos=巫n 1(3) cos =-(4) cos= 0 )(5) cos = -l

10、623 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 + b) = cos4cossin asin bcos(a- b) = cos 4 cos + sin a sin btan(a 一 b)=tan

11、 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 /1-cosasm = j2*2acos=2fl + cosav 2a /l-cos4 tau = j2 v 1 + cos asin a1 +

12、 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)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-cot