微分積分公式大全

1、高等數(shù)學微分和積分數(shù)學公式（集錦）（精心總結(jié)）nn 二a0X + a1XmAlimanxb0xm +曲二1 +bm30b0m（系數(shù)不為0的情況）二、重要公式（1）lim0sin x=1(4) lim=1n_lim(2) lim (VHx)兀arcta n x =21|x = e(3) lim 習(a0)=1兀(6) lim arc tanx = 1產(chǎn)2(7) lim arccot x =0X平(8) lim arccotx s(9) lim ex = 0x_oC(10) lim ex =處(11) limX =1T十下列常用等價無窮小關系(XT 0)sinX : X ta n X : X ar

2、cs in x : x arcta n x ： x 1 cosx :In (1 +x ) ： X eX -1 ： X aX 1 : xln a (1 +x f-1 ：ex四、導數(shù)的四則運算法則(uv)=UV(uvj=uV+uHU川3Iv丿五、基本導數(shù)公式fL!Llf(c ) =0 X = Ax (sin X ) = cosx(cosx ) = -sin X(tan x ) =sec x(6)( cot x )=2-csc X(secx ) =secx 4an x (esex ) = -cscx cot xr 1 gx) = ex(10) (ax ) = ax In a(ii) (In x)=X

3、X 1-1-(12) (log a ) =(13) (arcsin x ) =(arccosx )=X l nav1-x21-X211p1(I5)(arcta nxj=；(arccot x j =(17)( X) =1(18)(V x) = 六、高階導數(shù)的運算法則(1) u(x)v(x)F)= u(x)(n)v(x(2) cu(x)()=cu(n)(x)u (ax+b)F)=anu(n )(ax+b )( 4) u (x)v(x 艸)=送 cjuL* x v(k) ( x)k=0七、基本初等函數(shù)的n階導數(shù)公式(1)(xn)(nLn!(2) (ea2)CLanea(aX)()=aX|nnaL心

4、n 兀)(4) |sin(ax+ b)j= a sin (ax +b+n”一丿T卩 nf兀I(5) Lcos(ax+bJ=a cos(ax+b+ n n I“n a心T )一卄(ax +b )in (ax +b)=(-1 jn_1 a(n -1 fn(ax + b )八、微分公式與微分運算法則 d(C )=0 d (xP)= Axdx d (sinx ) = cosxdx2 2 d (cosx )= -sinxdx d (tanx )=sec xdx d (cotx)= -csc xdx d (secx )=secx tanxdx d (cscx )= -cscx cot xdx1 d (e

5、= exdxd (a=ax in adx(ii) d( ln x ) = - dxx1 1(12) d (log aX ) =dx (13) d (arcs in x)= =dx d (arccos x )=-xhaJ1 X21 11 +x5 d (arctan x ) = dx( d (arccot x ) = 2 dx 1 +x九、微分運算法則 d(U v )=du dv d (cu )=cduv2 d (uv)=vdu +udv d (叢=皿 udvIv丿一、下列常用湊微分公式積分型換元公式1ff (ax +b dx = = f f (ax +b d (ax +b ) a、u = ax

6、+ bJf (xXdx 二1 Jf du =xff(l nx)丄 dx= ff(l nxd(l nx)、xu = I n XJf (ex )exdx = Jf (ex p (ex)Xu =eJf (a )a dx -Jf (a d(a )In aXu = aJf (sin X ) cosxdx = J f (sin x d (sin x )u = sin XJf (cosx ) sin xdx = - J f (cosx d (cosx )u= cosxf f (tan X ) sec xdx = J f (tan x d (tan x )u = ta n X2Jf (cot X ) csc

7、xdx = J f (cot X d (cot X )u = cot X1f f (arctan x ) dx - f f (arcta n x d (arcta n x )、1+xu = arcta n x1f f (arcs in x dx - f f (arcs in x d (arcs in x )訥-X2u = arcs in x十、基本積分公式jkdx =kx +c Jxx = x+c = In X +c 4+1、xaXdx=+c rexdex +c rcosxdx =sinx+c In a 1fsin xdx = cosx +c fdx 、 cos x=J sec xdx = t

8、an x + c(11)sin x2 1=J CSC xdx = cot X + c(10)J7 dx = arcta n x + c1+x2f 1 dx = arcsin x + c十二、補充下面幾個積分公式ftan xdx =In cosx +c Jcotxdx = In sin x + cfsecxdx = In secx + tanx +c Jcscxdx= In cscxcotx +cF 1,1Xf 2 dx = arcta n + ca +x a a宀xInX -a 2aX -aX +adx=arcsinSc a2-X十三、分部積分法公式形如fxneaxdx，令 un=X形如fxn

9、 sin xdx 令 u形如形如形如形如十四、X + Jx2 a2 +cdv = eaxdx=xn， dv =sin xdxJ xn cos xdx 令 u = xn， dv = cos xdxfxn arctan xdx，令 u = arctan x， dv = xfxn In xdx，令 u = In X，dv = xndxndxfeax sin xdx， feax cosxdx 令 u= eax,sin x,cosx 均可。第二換元積分法中的三角換元公式(1) Ja2 -x2 X =as int(2)J a2 + x2 x=ata nt(3) Jx2 -a2 x = asect【特殊角的

10、三角函數(shù)值】(1)sin 0=0(2) sin丄6 2.兀73 sin = 32(4)JIsin 一 =1)2(5)si n 兀=0(1)cos0 =1(2) cos二6 2(3)兀 1cos=-32(5)COS = -1(1)tan 0 = 0(2)tant6(3)tanr兀tan 不存在2(5)tan兀=0十五、cot 0不存在兀(2)cot6JIcot = 02（5）cot兀不存三角函數(shù)公式1. 兩角和公式sin( A + B) =sin AcosB +cosAsin B sin(A -B) =sin AcosB -cosAsin B cos(A + B) =cosAcosB sin A

11、sin B cos(A B) =cosAcosB +sin Asin B丄 z A f tanA + tanB 丄 “ tan A-tanBtan(A + B) =tan(A - B)=1 -ta nAta nB1 + ta nAta nBcot B +cot A一“ c、cot A cot B-1 一“ c、cotA cotB+1cot( A + B) =cot( A - B)= f -cot B cot A2. 二倍角公式2 2 2 2si n2A=2si n Acos A cos2 A =cos A-si n A=1-2s in A = 2cos A 1tan2A= 2tanA1 -ta

12、n1 2 A3. 半角公式1 -cosA A +cosA co込二 jtanA2(1 -cosA VVcosAsin AcotAjrod sinA1 +cosA2 Vl-cosA 1-cosA4. 和差化積公式a +ba +b5 2 a -bsin a +sin b =2sintos:2 2a -b coscosa - cosb = 2sin2 2 2cosa +cosb =2cosa +bsina-sin b = 2cos2a + bsin 口2.a -bsn2sin (a +b )cos a cosbtana +tan b =5. 積化和差公式1 Lcosacosb =- cos(a + b )+cos(a - b)1 嚴1-sin acosb =5 sin (a +b )+ sin (a -b )cosasin bsin(a + b )sin (a -b6. 萬能公式2ta na2sin a =厶1 +ta n2 旦2彳丄2 a1-ta n cosa =.齊次微分方程：労= tana1+ta n2 22ta na21 - tan2 a2sin2 X +cos2 X =1sec xta n2 X = 1CSC2X - cot2 X = 17. 平方關系8. 倒數(shù)關系tanx cotx=1secx cosx=1 cscx s