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1、常用积分公式(一)含有axb的积分(a0)1. x-=-lnaxbCaxba2. (axb)dx=(axb)1C(1)a(1)C3.x.1,-dx=(axaxbabblnaxb)C4.2xaxb1(axb)222b(axb)b2lnaxb5.dxx(axb)1lnaxbb|x6.dxx2(axb)-1%nbxbaxb7.x.1八2dx=r(1n(axb)2a2ax8.x21-2dx=(ax(axb)2a39.dx2x(axb)b(axb)axb2blnaxb51naxbb2axb(二)含有-、axb的积分10. Vax_bdx=-7(ax_b)3C3a11. x7axbdx=2-2(3ax2b
2、)s/(ax_b)3C15a12. x2Vax_bdx=-2-(15a2x212abx8b2)V(ax_b)105a13., ax bdx=-22(ax2b)VOxbC3a214.2xax b-3(3a2x24abx8b2)、,ax15a315.dxx.axb1bln16.dxx2axb17.、axbjbaxb.bC(b0)arctanC(b0)bx2bx,axbax-b-dx=2/axbxdxxraxbxZbdx = xa dx2 x、ax b20.dxx,22、n =/2,22.n 1(x a )2( n 1)a (x a )2n 3-22(n 1)adx72 n 1(x a )18.(三
3、)含有x2a2的积分dx1x19. 2=arctanCxaaadx1,xa22=lnxa2axa(四)含有ax2b(a0)的积分22.dx2ax bx12一23.一2dx=lnaxbCaxb2a2#.25.26.27.28.29.30.(六)32.33.34.x2axb,xdx=adxx(ax2b)dxx2(ax2b)dxx3(ax2b)dx22(ax2b)2含有ax2dxaxb12bIn1bx2x2ax-2ln2b2dx2-ax2axb2x22b(ax2b)12bdx2axbbxdx2axbxcc(a0)的积分24ac=arctan2axb二Cb24acb22axb、b24ac(b2.b24
4、acx.12-dx=axbxc2a含有，x2dx22、.xadx.(x22axi2ulnaxbxbb24ac2aax(b24ac)4ac)2/a(a,xarsh-a0)的积分C1=ln(x2X3a)-2-Caxa=dx=Vx2a2C2a.(x2a2)3dx=Cdx2bxcx2a2)C#.,x2a22X-22a/dx=vxaln(x2236.37.38.39.40.41. .42.43.44.2xdx=x./(xCTJn02Qcx.x2a2ax2adx=x2a2ln(xVx2a2)C2272T3,x22、23422、v(xa)dx=-(2x5a)vxa-aln(xVxa)88xjTadx=-7(
5、7a2)3C342-22.x22、22a22._xMxadx=-(2xa)vxaln(xxxa)C88-22：22xa“-2,.xaadx=vxaaIn-Cxxxxln(x.x2a2)C(七)含有Jx2a2(a0)的积分x46.archaC1=lnxC47.dx.(x2a2)3x77Tdx=x2a2C#.49.50.52.53.55.56.57.58.59.60.2x2x2xdx=dx=2X3a)dx=2-In2=dx=23a)Inx2axx2a2dx222x.xa1=arccosa22xa2ax2-In2x2a2a/22、3,x(xa)dx=(2x854.xx/x2a_2dx=3.2x5a2
6、),x2lna2)3C.x2adx=一(2x822xadx=x22xa2dx=x含有a2dx_alx2dx.(a2x2)3a2).x2a24-ln8aaarccos一xInx(a0)的积分arcsinCax-22,dx=Jax22ax62. x=dx.(a* 2 x2)363.2 a . x arcsin 一 C2x.65.66.67.69.70.64.x.a2x2dx1,t=lnx工a2x2adx2-22xaxarcsin-Ca22ax2axxdx=2a一xarcsin一(a2x2)3dx=-x(5a22x2)Ja268.34x一aarcsin一xG7dx=3府x2)3C2-22x2xvax
7、dx=-(2x82、22a)-ax4a一x一arcsin一22ax,2dx=vaxx22a、.axalnCx72.ax2dx=x,alarcsin-Ca(九)含有aax2bxc(a0)的积分73.2a Tax2bxc2dx=；ln2axbvaxbxc7a#.ax bx=2axb2=cdx=axbxc4a75.,ax2bx76.dxcbxax277.78.4acb28mIn2axb2VaVax2bxcCdx=Tax2bxcab=ln2axb2逝启2好12axb-arcsin二Ca,b4acx2bxcVcbxax2dx=2axb2b24ac.2axb-ccbxaxarcsinC4a8,a3.b24
8、acx,=2=dx=.cbxax21TT_7b_2axb-cbxax一arcsinCa2,a3b24ac(十)含有或(xa)(bx)的积分80.=(x79.a)ln(J7|xb|)CJ3dx=(x:bxb)H(ba)arcsinJ(xa：(bxVn后C(ab)82.(xa)(bx)dx =2x a b7(xa)(b_xj -a)-arcsin JX a C 4. b x(ab)(十一)含有三角函数的积分83.sinxdx=cosxC84.cosxdx=sinxC85.tanxdx=IncosxC86.cotxdx=Insinx87.secxdx=Intan(4x、八,一一)C=lnsecxta
9、nx288.cscxdx=In,xtan2C=IncscxcotxC89.2secxdx=tanxC90.csc2xdx=cotxCsecxtanxdx=secxC92.cscxcotxdx=cscx93.2.xsinxdx=一294.27xcosxdx=一1-sin2x41.。-sin2x95.96.24n1n1sinxdx=sinxcosxnn1n1.cosxdx=cosxsinx97.98.dxsinnxdxcosx99.n1n1sinxsinxnncosxn1cosmncosxsinxdx=mn2sinxdxn2,cosxdxdxn2sinxdxn2n1cosxm1,n1m1cosxsinxnmn1m1.n1ncosxsinxmcos2.nxsinxdxm_n2cosxsinxdx103.dxa bsin x2,a2=arctan b2atan-2,a2 b2bC(a2b2)104.dxa bsin x一 In2 a,x ,a tan b2b2 a2(a2b2)105.dxa bcosx106.atan2b ； b2 a2-J-b arctan(a b . x.tan-)( a b 2(a2b2)dxa bcosx,x tan2,x tan-2C2 2,2 x(a b )100.sinaxcosbxdx=
