常用積分表完整版

1、常用積分表完整版常用積分公式dxx(ax b)ax（一-）含有axb的積分（a0)dx1ax bC1.-Inbaxa2.(axb)dx1(axa( 1)3.xdx1-(ax bbln2axba2xdx11 ,4.-(axb)2dx3axba、2b5.b) 1 C (1)ax b) C2b(ax b) b2ln ax b C6.dxx2(ax b)1bxax b7.1=-yQnaax2x(ax b)jdx9.2bln ax bb2ax b) Cdx2x(ax b) b(ax b)ax b（二）含有.ax b的積分10.11.12.13.14. 2Jax bdx = J(ax b)3 C 3ax、

2、._ dx =4(3ax 2b) (ax b)3 C15a2x2 一 ax bdx =(15a2x2 12abx 8b2) (ax b)3 C105a-dx b2b) . ax b2荷(3a2x2 4abx 8b2) b C1 lnax b bdxx * ax bdxx2 . ax b.ax b2 dx = x含有x2dx 2x a_ dx (x2dx _2 x含有VbTaxb y/b2 arcta n、 bax bbx2、ax bax b2a的積分=arcta n a a2n2/2a )2(n1)a (xa2ax2dxax2 bxax2x2ax bax bC (b 0)C (b 0)|xal

3、xalC丄ln2ab(a 0)的積分a2bdxx、ax bdxdxb2、n 1a )2n 32(n 1)a2dx/ 22、n 1(x a )a ax b1a carcta n x C ab一1In12adx =adxx(ax2 b)(b 0)C (b 0)ln ax2 bln 2bdxax2 b2x2axdxx2(ax2 b) bx bdxax2 b15.16.17.18.(三)19.20.21 .(四)22.23.24.25.26.27.28.(五)29.30.(六)31.32.33.34.35.36.37.38.dxax2 bx c22ax b礦b2翻*門扃6 C1.b2 4acln2ax

4、 b 、. b2 4ac2ax b - b2 4ac2(b 4ac)2(b 4ac)dxax bx cIn ax2bx c -dx2a ax bx c含有x2 a2 (a 0)的積分dxarsh- C1 = ln(xx2 a2) Cadxx1xdx2、3a )x2dx =a2 x2a2)x2dx =ln(x.x2a2)dxIln a2 2vx .a a C2 .dxa ax b 122 ln 23“2.、 2 ln-22 (x (ax b)2bx2bxdx x1dx2 2(ax b)2b(ax2b)2b2 , ax b含有ax2 bx c (a 0)的積分dxa2x8a4|n(x r)C39.

5、40.41 .42.43.44.(七)45.46.47.48.49.50.51 .52.(xX dx = =2 ,(x2 a2)3、x a 222 a a In a2)3dx = (2x2 5a2).x2 a28xadx = 1 /x23*2 2. x 2x x a dx =(2x8x22 ax.x2a2x2dx = . x2 a22 2dx =三丄 ln(x2 2 a Inxa x2a2)含有x2 a2 (a 0)的積分dx x-22.x aarch G=ln aa2dx,(x2 a2)3xdx =2 2- x2 a2.x a2門vdx=In2 av x2a2dx1=arccosx、x2 a

6、2 adx-22x a2a x222 x 22 a _x a dx = Vx a ln x2 2ln x Vx a.(xx dx =2 2 ,a x =2 C a2)3dx = -(2x2 5a2)、, x2 a8adx = /(x23*53.54.55.56.57.58.(八)59.60.61.62.63.64.65.66.2 2. x 2x x a dx =(2x8a2)、疋4ln822xx a2 2x a . dx =xVx2 a2 aarccos 各lxlIn xa2.x2 a2x2(a 0)的積分dx.a2 x2=arcsin C adxx 22、32 22.(a x ) a .a

7、xdx.(a2 x2)3122.a x22arcs in C2a2,(07dx=arcsin - Car22a a xCdxa2 x2 小. x .-.7.a x dx = a2(a2 x22a . xarcs in2a3 ,x 22、 )dx =(5a 2x ) . a2 2. x 2x a x dx =(2x8a2x23a4arcsi n C8xv Ca2) a4a . xarcs in8a-2 dx = . a2 x2a2 x2aln aJa_x. C.x arcs ina含有ax2 bx c (a 0)的積分dx.ax2 bx c ax2 bx cdxIn 2ax b 2掐Jax2 b

8、x c2ax-faixbxc4a4ac b28 . a3In 2ax b 2需 bxc Cdxc bx ax2i c bx ax2 dxba32 ax2- a ax2bxtarcsin_2ax_b_b2 4acdx =- c bx ax2含有2ax b飛盂4ab2 4ac8 . a3arcsin2ax b,b2 4acbx ax2b . 2ax b =arcs in:-_a或a)(bx)的積分x b 67.68.69.70.71 .72.(九)73.74.75.76.77.78.(十)打 bdx =(X=2arcsin Jx a.(x a)(b(b(b方dx = 2x aa)l n(v|x b

9、|) Ca)arcs inx ab x(a b)_J(x a)(b x)arcsin4b x79.80.81.82.(a(十83.84.85.86.87.88.89.90.91.92.93.94.95.b)含有三角函數(shù)的積分sin xdx = cosx Ccosxdx = sinx Ctan xdx =In cosxcot xdx =lnsi nxsecxdx =lntan (4x、-)C2=In secx tanxcsc xdx =lntan 2C = lncscx cotx Csec xdx = tan x C2csc xdx = cotx Csecx tan xdx = secx Ccs

10、cxcotxdx = cscx Csin 2x4sin2x4xsin xdx =2x cos xdx =2sinn xdx =1 n 1sin xcosx nn 1. n 2,sin xdx nn .1n 1_n 1n 2.cos xdxcos xsinxcos xdxnndx 1cosxn 2dxn1 sin xn 1n 2sin xnsin xdx 1sin xn 2dxn1n 1n 1n 2cos xncos xcos x96.97.98.99.100.101.102.103.104.105.106.107.10 8.109.110.m. n1ecu v o i n vH v m 1.

11、n 1m1m 2. necuv qinvHvUUo A oil 1 AU A匕on人nil i人UUoX ol 11AUXmnmn1m 1. n 1n1m.n 2. cos xsinxcos xsinxdxmnmnsin ax cosbxdx =12(acos(ab)b)xcos(ab)b)x Csin ax sin bxdx =1sin(a b)x2(a b)1sin (a b)x C2(a b)cosax cosbxdx =1 sin(a b)x2(a b)dxa bsi nxata n?arcta n 1 2 a2 b2:a2b2b廠 C (a2b2)dxa bsi nxatab Vb2

12、a22atan* b Vb2a22(a2b2)dxa bcosx-_- arctan( J _b tan)a b 2(a2b2)dxa bcosxab aa b(aCCabdx2 2 722a cos x b sin x1 arcta n( tanx) abadxa2 cos2 x b2 sin2 x丄ln2abbtanx abtanx a1-x cos ax C a2 .x sin ax a23 cos ax axsin axdx = A sin axa2 1 2x sin axdx = x cosaxa, 1 1xcosaxdx = - cosax xsinax Caa2 , 1 2 .x

13、 cosaxdx = x sin axa含有反三角函數(shù)的積分(其中22 x cos ax aa 0)23 sin ax C aarcsin -dx = xarcsin . /a-2xxarcsi dx ax?a?()arcs in242x2 C32. x t x . xx arcs in dx = arcs ina1 ( 29(x2a2) .a2arccos-dx =axxarccos-a2x arccosdx =( a 22ax)arccos 一4ax-22:a x432X, xx arccosdx = arccos a 3a1(x2 2a2) . aarctan dx = x arcta

14、n ?ln (a2a-x2) Cx arcta n x dx = 1 (a2 a 2x2)arcta n -a3x2 arcta ndx =arcta n a 33a 2ln( a6x2)含有指數(shù)函數(shù)的積分1a xdx =ax CIn aax1 axe dx = e C axe dx = 2 (axan ax1 n axx e dx = x eaax1)en 1 ax .x e dxxaxdx =x x a ln a(In a)2a Cn x1 nx a dx = x aIn ax nn 1 x .x a dxIn aeaxsin bxdx = 21 2eax(asin bx bcosbx)1

15、11.112.(十二)113.114.115.116.117.118.119.120.121.(十三)122.123.124.125.126.127.128.a b129.eax cosbxdx =2 eax(bs inbxa ba cosbx)ax n e sinbxdx =ax n 1e sinbx(asinbxa bn2n(n 1)b2 ak2 2 b n131.ax e cosn(n1)b222 2ab n130.（十四）ax n 2e cos bxdxn bxdx = -_1 2 2a bnax n 2e sin bxdxax ecosn 1 bx(a cosbxCnbcosbx)n

16、bsin bx)132.In xdx = xlnxxCdx,133.=lnln xCxln xn -1n1八1134.x ln xdx =x(lnxn1n135.(ln x)ndx =x(ln x)nn(ln x)136.xm(ln x)ndx =1m x1r(ln x)含有對(duì)數(shù)函數(shù)的積分nm1)C1dx亠 xm(lnm 1x)n 1dx（十五）含有雙曲函數(shù)的積分137.shxdx = chxC13 8.ch xdx = shxC139.thxdx = lnchx:C140.sh2xdx =-1sh2x C24141.ch2xdx =1sh2x C24（十六）定積分cosnxdx =142.s

17、in nxdx = 0143.cosmxsin nxdx = 00, m cos mx cos nxdx =145.sin mxsin nxdx =0, m n146.o sin mxsin nxdx = ocosmxcosnxdx =0, m n,m n 2147. = 2 sinn xdx = 2 cosn xdx00-n 1I1 n 2nin 1 n3 II4 2n n2 H15 3n11 n3III1 nn n2（為大于1的正奇數(shù)），=13 1（為正偶數(shù)），=4 2 22換元積分法、第一換元積分法（湊微分法）貳何Q輕辦=J妙血=玖+C= 7x） + C、常用湊微分公式昭類型換元公式第

18、換元積法I =召 Jf （的十時(shí)dgjr 桶）（0J4”）廣【礦）疔 d ”陽(yáng)）說(shuō)滋=右Jf（”）姑 Jy（sinJt） -ccsxJj = Jj（sinJt）Jsinjc 7.Jy（cos） siiix= -J/coexXeas x J/（tanx）E8c2 xdx - J/ftaniitJ tanx .Jcatxjcsc2 Xtiz = - Jf （cot Rd ent * 10 Jy（3rctani） 1JXarctanjc）貞騏仙工）II f/（arcsinj） 】_（aicsin7）rf（anc9mx）Jl - Fu - ax +iU -lhXu =,1? = 43li =siiirUu fauxU =COtY社=an;tan.A