微积分常用公式教程文件

1、此文档仅供收集于网络，如有侵权请联系网站删除只供学习与交流1.常用等价无穷小当 x 0 时 x sin x tan x arcsinxarctanxln(1 +x)ax-1(1+x)b-1In a(其中a 0, ?工0)2.常用极限1.3.5.7.9.11.13.15.16.18.20.22.24.26.28.29.30.31.32.xexknlim 歹=0xs a,(alim nqn ,(|q| 1)1)a (1+ x) a2.4.6.limXsn c肓=0,(c 0)nlim a = 1,(a 0)xslim也xsn0,(a 1)nn7H!= elim竺x +s x=0 ,(a 1,? 0

2、)8.10.12.1lim (1 + -)xsnsin xlim =xs x”1p+2 p +?+nlim (xsnP+1=丄丿 p+11p+2p+?+n pxm(严1lim (xs n+11+ +n+2xmxoxoxolimx1x1P+T)=1+ 2n)14.lim (xs1 p+3 p +?+(2n-1)np+1p-)2pp+1=ln2sin xxax-1xln (1+x)xarcta nxxlnnv!+ ax - V 1+ Bxa Bm-;,(mn 工0)17.19.21.23.25.xo(1xolimx0limx0xo(1+a)1x) = earcs inx,=1x(1+mx) n-(

3、1+nx)x2mn ,V1+ ax ?V 1+ B出1x=-mn(n - m)a B /+, (mnm nxm-1xn-1m /=J(m,n为自然数)27.limx 1n _ m-n1-x n) =2n*1 = m,(m,n *Z)若Xn (n=1,2)收敛，则算数平均值的序列1? -(X1 + X2n? Xn),(n = 1,2 ?)也收敛，且x1+x2+?+xnlimxsnlim xnxs若序列 Xn(n=1,2)收敛，且 Xn0贝UJ“1 + x2 + ? + xn =Xn+1右 Xn0(n=1,2)且xs 百存在，则lim VXn = lim 弓匕xsxs Xnlim Xnxs若整序变

4、量Yn +s，并且一一至少是从某一项开始一一在n增大时Yn亦增大,Yn+1Yn 则XnXn-Xn-1ns Yn= ns YnYn?此文档仅供收集于网络，如有侵权请联系网站删除3. 常用公式及不等式3.13+ 23+ ?+n3=(1+2+?n)24.a3 b3 = (a+ b)(a2 ? ab + b2)5.xn-1 =(x-1)(xn-1 +Xn-2+ ? + x +1)6.xn-an=(x-a) (xn-1 +axn-2 + a2xn-3+ ? +a2x+ an-1 )7.xn+ an=(x +a) (x 2k-1-ax2k-2 ) + ?+ (a2k-2x -a2k-1 )8.x -1 =

5、(Vxn-1+ Vx-2+ ? + 1)1.1 + 2 + n =n(n+1)22.12 + 22 + ? + n2n(n+1 )(2 n+1)69. 伯努利不等式(1 + x)n 1 + nx(1 + x1)(1 + x2) ?1 + xn) 1 + x1 + x2 + ? + xn11. |xy| xy10. |x- y| |x|-|y|12.|X+ X1 + ? + Xn| |X| -(|X1| + ? + |Xn|)13.n! n+1(丁)14.1 32n-1?- ? ?-2 V2n1V2n+11 1 115.- ln(1 + -) -n+1n， nn a-117. va - 1 nA

6、 k!18. 组合数公式Cn =计=knX16.1 + a 0)va-x 2a2. /1 dx = x + c14. dx = In 凶 + c.16. /2 dx = arcsinx + cv 1-x28. /sin xdx = - cosx+ c110. /-厂 dx = - cot x+ csin 2 x12. /shxdx = chx+ c114. /-厂 dx = -cth x+ csh2 x厂 dx1x16/ _2 = arctan + c,(a 工 0)ax+x2aaxdx1 o o18./ax7X = 土訓 |a2 + x2| + c20./- d2x 2 = In |x +

7、Vx2 + a2| + cV(2 a221. = VaF +cx+ c, (a 0)22./Vx2 a2dx = 2 Vx2 a2a2 2In |x + Vx2 a2| + c22./Va - xsin 9?csc 9= 13. tan 9 ?cot 9 = 15. sec2 9- tan 2 9 = 1sin 97. tan 9 =-cos 9 cos 2 a = cos2 a- sin2 a= 2 cos2 a- 1dx = 2 va - x2 + arcsin2只供学习与交流7.三角学公式sin 2 9 cos2 91. 基本关系2. cos 9?sec 9 = 14. sin2 9 +

8、 cos2 9 = 16. csc2 9- cot2 9 = 1cos 98. cot 9 =-sin 92. cos( a p) = cos acos p? sin asin p3.tan( a p)=tan a tanp1?tan atan p4.cot ( a p)=cot a cot p ?1cot p 士 cot a2. 两角和与差的三角函数公式1. sin( a p) = sin acos p cos asin p1 - 2sin2 a =1-tan 2 a1+ta n 2 a3. 倍角公式2 tan a1. sin 2 a = 2 sin acos a= 厂1+tan a3.ta

9、n 2 a =2 ta na1-ta n2a5.sin 3 a =3 sina-4 sin3 a4.半角公式1.2 a1-cosa2.sin2=222 atan 2 =1-cosa1-cos asin a=( )2 = ( )2 sin a1+cos a3.1+cosa4.cot 2 a =2 cot a -12 cot a6. cos 3 a= 4 cos3 a- 3COS a2 a 1+cos acos2-4.2 a 1+COS acot2 21-COS a1+COS a 2/ sin a o()=( ) sin a1-cos a5和差化积公式1.sin a+ sin2sina + 3co

10、s2.sin a-sin2cos sin23.a + 3a - 3cos a + cos 3= 2 cos 丁 COS 丁4.cos a -a + 3cos 3= -2 sin 丁 sin5.tana tan 3 =6.cot7.tana cot 3 =sina sin 3cos(a ? 3)sina sin 3cos(a ? 3)cos a sin 3a cot 3 =si n (a36.积化和差公式1.sin2.cos3.sin1asin 3 = - 2cos( a+ 3)-1 ,a cos 3 = 2 cos ( a+1 /a cos 3=2 sin ( a +cos( a- 3 )3) + cos( a 3 )3) + sin( a 3 )7.双曲函数的基本关系1.3.cosh21- sinh21 = 12 1coth21 = 1 +sinh t2.4.2 11 - tanh21 =27cosh t2sinh 2x = 2 sinh x cosh x = 2 cosh x ?tanh x5.ex-e -xsinh x = 26.x -x e +ecosh x =27. 双曲余弦的反函数8.万能公式x =