常用積分表知識分享

1、（一）含有ax b的積分（a0)1.axdx 1|nb aax2.(axb) dx(axb)3.dxax b12 (ax b ablnax4.2x .dxax bb)25.dxx( ax b)1lnbax6.dxx2(ax b)1bx7.x2dx (ax b)21=-r(lna2x(ax b)2dx9.10.11.12.13.14.axb) C2b(ax b)ax bbax b)b2lnax=(ax2bln ax bb2ax b)dx2x(ax b) b(ax b)含有ax b的積分12 In b2Tax dx = 7(axb)33a22(3ax 2b15ax、ax bdx =x2 . ax

2、bdxXdx ax b2J dxax bax b).(ax b)3 CJ(15a2x2 12abx 8b2) ., (ax b)3 C105a7(ax 2b)、ax b C3a2芻(3a2x2 4abx 8b2). ax b C 15a15.dxx Y ax b1 ln Jax b Jb 品 Vaxb VbC (b 0)16.dxx2 ax b17.18.ax b2'x含有x219.dx2a20.(x221 .(四)22.23.24.25.26.2arcta n bax bbx2 ax bax ba2的積分1x=_ arcta n _ aaax ba2bC (b 0)dxx , ax

3、bdxx, ax bdxx、axdx2、n_c/八 2/2a )2(n1)a (x2、n 1a )2n 32(n 1)a2dx22、n 1(x a )dx2 x i含有2 axax2dxbx2ax-dx b2x2-ax b|xalxalC丄ln2ab(a 0)的積分arcta njax C .ab2j abTaxdx =dxx(ax2 b)(b 0)C (b 0)12aln ax2 bln 2bdxax2 b2x2axdxx2(ax2 b) bx bdxax2 b27.2ax b2x28.dx2 2(ax b)22b(ax b)12bdxax2 b29.30.(六)31.32.33.34.35

4、.36.37.39.含有ax2bxdxax2 bx c2-dxax bx c含有.x2dx, x2 a2dxc(a 0)的積分,ln. b2 4acln ax 2a2 /a (aarsh- apdx =adx2、3a )2x2 2 x adx =dx23 a )dx2 ： 2 x x a、x2 a2dx丄 2 ax b carcta nC2ax b b2 4ac2(b 4ac)(b2 4ac)2ax4ac2 bx0)的積分C1 = In (xxa2 一x2 a2,x2a2CxF2dx =llna2aIn(x2b2adxax2bx ca2) C、x2a2)xr22x aln(xx2 a2)、x2

5、a2 aC2 2.x a2a x2a . /In(x、x2 a2)3 4a ln( x x a ) C840.41 .42.43.44.(七)45.46.47.48.49.50.51 .52.53.54.55.:x 222、(x a ) dx =(2x 5a ) x8a2)3 Cx x =-3a2) 、 x2 a22 2. x 2x x a dx =(2x84±ln(x.x2 a2) C8x22 ax.x2a22xdx =dx =含有.x2.x2 a2.x2a2dxln(x2a (a 0)的積分、x2 a2) arch兇 C1 =lnx Jx2 a2xaCdx汙22x a,(x2 a

6、2)3a2x；=a2 dx =x2 xdx = J 212.x adx =2a .ln2=ln2 a、x2a2a =arccos a xdx2 22x x a22.x a2a xx2 a2dx =2x2aIn2(x2 a2)3dx = -(2x8r22 ixx a dx =5a2 a3a4ln8r22x x ax2a2)3C56.X2 Cdx =評X2 a2)、C4 In8xx257.22.x a22adx = -. x a a arccos x x58.x2a2x2In x59.60.61.62.63.64.65.66.含有,a2 x2 (a 0)的積分dx：=2 2a xdx=arcs i

7、n? C a.(a2 x2)3xdx =2 2 a x12 2a xdx(a2 x2)32t x dx =；2 2a x2X x:=dx =-(a2 x2)3. a x2a . xarcs in2aarcsinx Cadx1 , a a2 x2= ln x i a2 x2 adxx2a?22a x2a x67.；a2x2dx =68.(a2x2)3dx69.x、a2x2dx =70.2 2 x ax2dx2a . xarcs in2ax 2 c 2、22=-(5a 2x ) - a x81x2)3C評x22、 2 2 a )' a x3 4. xa arcs in8a4a . x ar

8、cs inC2 2旦亠dxa2 X2aJa2 x2a l n xx含有ax2 bx c (a 0)的積分dxax2 bx c、ax2 bx cdxxax2 bx=dxcdx.c bx ax2、.c ax lxx 2dx.c bx ax2含有x :dx=(x=(xIn 2ax b 2品bxc C2ax4a色n 2axln 2ax b2疑b 2 a jax2 bx2、a、ax2 bx c1. 2ax barcs in a. b2 4ac2ax b ,2c bx ax 4a或、.(x a)(bdx.(x a)(b x)b2 4ac .arcs in 8. a3、b22ax b2ax bb .:arc

9、s in 2a3. b2 4acx）的積分bJ b xx a)(b x)dx =(b a)l n( (b=2arcs inx aL C (a b)2x aC4aca)arcs inC2b(b a) .(x a)(b x)arcs in71 .72.(九)73.74.75.76.77.78.(十)79.80.81 .82.）含有三角函數的積分(a b)sin xdx = cosx Ccosxdx = sinx Ccot xdx = Insi nxCsec xdx = Inta n(一 -)C4 2csc xdx = Intan C=In22sec xdx = tanxCcsc2 xdx =cot

10、xCCIn cosxtan xdx =In secx tanxcscxsecx ta nxdx = secx Ccsc x cot xdx = cscx2x 1sin xdx = -sin2x247x 1cos xdx = sin2x24n1 n 1sin xdx = sin xcosxnn |1 n 1.cos xdx = cos xsinx n1 cosxn 11 sin xsin xdx.nsin xdxn.cos x n 1m n i cos xsin xdxn 1 cos x1cotx Cn 2sin xdxcosn 2 xdxdx.n 2sin xdx彳 n 2n 1 cos xm

11、 1. n 1m 1cos xsi nx m nm n1m 1n 1cos xsi n xm nsin ax cosbxdx =cos(a b)x2(a b)mcos2. n .xsin xdxm n 2|cos xsin xdx m ncos(a b)x C2(a b)83.84.85.86.87.88.89.90.91.92.93.94.95.96.97.98.99.100.1sin ax sin bxdx =sin(a b)x2( a b)1cosax cosbxdx =sin(a b)x(a b)dxa bsinx2 a2b2ata n 仝arcta n =2一 b2.a2dxa bs

12、inx=2ln adxa bcosx1sin (a b)x C2(a b)1sin(a b)x2(a b)b2)atan2atan蘭 b 4ba22dxba bcosxdxa2 cos2 x b2 sin2 xdxa2 cos2 x b2 sin2 xxsin axdx =丄 sin axax2 sin axdx =b :b2 a2嗨）CC1 arctan( tan x) abaab丄ln2ab1 :x cosax abta nx abta nx a1-x cos ax C a2 .2 xs in ax ax cosaxdx = $ cosaxa1x cosaxdx = x sin axa含有

13、反三角函數的積分(其中1 . xsi n axa22 x cos ax aa 0)arcsin xdx = xarcsin 仝、a2 axarcsin - dxa2)arcsin432 x 1 x. xx arcs in dx = arcs ina 3(ax21 / 29(x(a2(a2b2)b2)arccosxdx = xarccos仝 aa23 cosax a2 .3 sin ax C a2a2).a2 x2 C101.102.103.104.105.106.107.10 8.109.110.111.112.(十二)113.114.115.116.117.118.119.120.121.(

14、十三)122.123.124.125.126.127.12 8.129.130.131.(十四)132.133.xarccosdx = a2(XT42ax)arccos 一4a32X Xxx arccosdx =arccosx2) Ca 3aarctan dx = x arcta n axarc叫dx=2(a2x2 )arcta nax C23自n(a2x2) C3x2 arctandx = arctan a 3含有指數函數的積分a xdx =ax CIn aax1 axe dx = e Caxeaxdx = 2 (ax 1)eax Caeaxs in bxdx =2eax(asin bxb2

15、bcosbx)eax cosbxdx =尹(bsinbxa cosbx)ax n e sin bxdx =Iax n 1-2亍。sin bx(a sinbxa bnCCn bcosbx)ax ne cos bxdx =2n(n 1)b2 , 2 2a bnax n e sin2 bxdxax n 1e cos bx(a cosbx a bnnbsin bx)2n(n 1)b2 . 2 2a bnax n e cos2 bxdx含有對數函數的積分ln xdx = xlnx x Cdx=In In xn ax .1n axnn 1 ax x e dx=-x exe dxaax x1xxa dx =

16、aaCln a(ln a)n x .1nxnn 1 x .x a dx =x ax a dxln aln axln x134.135.xn In xdx =xn 1(ln xn 1(In x)ndx = x(ln x)nn 1n (In x) dxm“、n |1 m 1“、nnm“、n 1 .136. x (In x) dx = x (In x)x (In x) dxm 1m 1(十五)含有雙曲函數的積分137. shxdx = chx C13 8. chxdx = shx C139. thxdx = In chx C2x 1140. sh xdx =sh2x C242 x 1141. ch

17、xdx = sh2x C24(十六)定積分142. cosnxdx = sin nxdx = 0143. cos mxs inn xdx = 00, m n144. cos mx cos nxd x =,m n0, m n145. sinmxsinn xdx =,m n0, m n146.sin mxsin nxdx =0cosmxcos nxdx =0,m n2n = 2 sinn xdx0n 1 ,=2 cosn xdx0n =11 n2nn 1n342nL -（n為大于1的正奇數），nn253n 1nn3L 3丄_（n為正偶數），I0 =nn242 22147.I1 = 1此文檔僅供收集

18、于網絡，如有侵權請聯(lián)系網站刪除換元積分法第一換元積分法（湊微分法）鈕侃;0 護僕伽= 曲伽=F（ti）+c= 7x> + Q二、常用湊微分公式換元會式積另類型uax-b u =15e 積 分 法S.Jy(smr) -cosxtix = ijjx池恥)S J/(tanr)sec? xdx - Jf (tatiRd 9.Jy(eot jt) csxdx = JRdjy (sinxsnu-s in 也=-j/cos A)(i cos xtaiLt0 =sin.rcoilu - tans$ =cot10 J(arctaiiM 廠打.血=Jy(aiEtamM(atctaiLi)11 J /(arcsin x) -'- Jjf = - j J (wsin j)d(aicginj)Jl - F配=aiutaniu = wesin x注：以上使用的多為三角代換,三角代換的目的是化掉根式 ，其一般規(guī)律如下當被積函數中含有a)b)c). >-.可令 一、_；. r. ，.可