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1、北大高等数学第3章习题解答习题3.1 求下列不定积分：111.1+2xdx=1+2xd(1+2x)=(1+2x)3/2+C.233x13322.2dx=d(x+1)=-+C.(x+1)22(x2+1)22(x2+1)113.x2x2+7dx=2x2+7d(2x2+7)=(2x2+7)3/2+C.4624.(2x3/2+1)2/3xdx=(2x3/2+1)2/3dx3/232113/22/33/23/25/3=(2x+1)d(2x+1)=(2x+1)+C.325e1/x5.2dx=-e1/xd(1/x)=-e1/x+C.xdxd(2-x)16.=-=(2-x)10099(2-x)99+C.(2-
2、x)1007.8.9.dx1dx1=3+5x231+(5/3)x23dx7-3x2=dx71-3/7x2=3d5/3x15=arctanx+C.51+5/3x2315177d3/7x13=arcsinx+C.2371-3/7x73dxdx=2=2arctanx+C.(1+x)x(1+x)ex11xx10.dx=de=arctane+C.22xx2+e22+(e)11.dxe-2x-1=dex1-(ex)2=arcsinex+C.dxdexdu11112.x-x=2x=-due-ee-1(u-1)(u+1)2u-1u+11u-11ex-1=ln+C=lnx+C.2u+12e+1lnlnxlnln
3、x113.dx=dlnx=lnlnxdlnlnx=(lnlnx)2+C.xlnxlnx2xddxdx2=-cot2x+C.14.=xx1+cosx22sin2sin222pdx+dx2xp15.=-cot2+C.p1-sinx241+cosx+2x141x101u2516.5dx=5dx=du(u=x5)444(x+1)5(x+1)5(u+1)1u2-1+11(v-1)2=du=dv(v=u+1)445(u+1)5v1v2-2v+11=dv=v-2-2v-3+v-4)dv(45v51111=-v-1+v-2-v-3+C=-(x5+1)-1+(x5+1)-2-(x5+1)-3+C.5353x2n
4、-11xn1u17.ndx=ndxn=du(u=xn)x-1nx-1nu-11111n=1+du=(u+ln|u-1|)+C=(x+ln|xn-1|)+C.nu-1nndxx4dx1du18.=(u=x5)555x(x+2)x(x+2)5u(u+2)111111u-du=ln|u|-ln|u+2|+C=ln+C.()52uu+21010u+2ln(x+1)-lnx1119.dx=(ln(x+1)-lnx)-dxx(x+1)xx+1=(ln(x+1)-lnx)d(lnx-ln(x+1)=-(ln(x+1)-lnx)d(ln(x+1)-lnx)1x+1=-ln2+C.2xearctanx+xln(
5、1+x2)earctanxxln(1+x2)20.dx=dx+dx2221+x1+x1+x1=earctanxdarctanx+ln(1+x2)dln(1+x2)21=earctanx+ln2(1+x2)+C.4112sin2xdsin2x=sin2x+C.24xxxx2x22.sin2cosdx=2sin2dsin=sin3+C.22223211123.sin5xsin6xdx=(cosx-cos11x)dx=sinx-sin11x+C.22112x-12x124.dx=dx-dx2221-x1-x1-xd(1-x2)=-arcsinx+C=-21-x2-arcsinx+C.1-x2x3+x
6、x3x25.dx=dx+dx2221-x1-x1-x21x21d(1-x)=dx2-21-x221-x221.sin2xcos2xdx=11122=1-xd(1-x)+dx2-1-x2221-x21=(1-x2)3/2-21-x2+C.3dx26.2(a0)23/2(a-x)x=asint,t(-p/2,p/2),dx=acostdt,(a2-x2)3/2=a3cos3t,dxdt1=dx=(a2-x2)3/2a2cos2ta2tant+C1x/ax=2+C=+C.2222a1-(x/a)aa-xx0, aa22=x-a-aarccos+C=x-a-p-aarccos+C-xxa=x2-a2+
7、aarccos+C.x22x2-a2dx=xy2-a2dy=yy2-a2-aarccosa+Cy27.x2-a2dx(a0).x0时,令x=asect,t(0,p/2).xdx=atantsectdt,x2-a2=atant,x2-a2dx=atan2tdt=a(sec2t-1)dt=a(tant-t)+C x22aax=a(sect-1-arccos)+C=a(-1-arccos)+Cxxaa=x2-a2-aarccos+C.x28.x2a2-x2a2-x2a2x1x=-arcsin-xa2-x2+a2arcsin+C2a2aa2x1=arcsin-xa2-x2+C.2a2dxe-3x/2d
8、x2de-3x/2229.=-=-ln(e-3x/2+1+e-3x)+C31+e-3x31+e3x1+e-3x22(1+e3x+1)(1+e3x-1)3x=-ln(1+1+e)+x+C=-ln+x+C3x331+e-12=ln(1+e3x-1)-x+C.3x31dx41du30.dx=(u=x4)41+x841+u21+x811=ln(u+1+u2)+C=ln(x4+1+x8)+C.44dx=-a2-x2dx+a2dx1dx-21u2du131.=-=-(u=2)2x41+x-221+uxx61+x2x71+x-21(v-1)21v2-2v+1=-1/2dv=-dv(v=1+u)1/22v2v
9、1=-(v3/2-2v1/2+v-1/2)dx231 1252222=-v-2v+2v253dxdx11211=-1+2+1+2-1+2+C5x3xx1+x=-5x5255232121+x+3x33231+x2-+C.x32.e2x31+exdx=ex1+exdex=ux33du(u=e)(u+1=v,u=v-1)31+uv5v2uv3-124=3du=3vdv=3(v-v)dv=3-+Cv1+u5233=(ex+1)5/3-(ex+1)2/3+C.5233.dx3+x-x2=dx113-x-+242=1dx-2131-x-42 2x-=arcsin12+C=arcsin2x-1+C.1313
10、22211291134.7+x-x2dx=7-x-+dx=-x-dx-2442212x-11291292+C =x-x-+arcsin224282922x-1292x-1=7+x-x2+arcsin+C.482935.dx,1+x-1=u,x=1+(u-1)2,dx=2(u-1)du,1+x-1dx2,1+x-1=u,x=1+(u-1),dx=2(u-1)du,1+x-1dx2(u-1)du=1+x-1u=2(u-lnu)+C=2(1+x-1)-ln(1+x-1)+C=2x-1-ln(1+x-1)+C.习题3.2 求下列不定积分：1x2121.xlnxdx=lnxdx=lnx-x2dlnx22
11、2x2121x21x2x2=lnx-xdx=lnx-xdx=lnx-+C.22x2224111122.x2eaxdx=x2deax=x2eax-eaxdx2=x2eax-xeaxdxaaaaa1212x2=x2eax-2xdeax=x2eax-2eax+2eaxdxaaaaa1212x2=x2eax-2xdeax=x2eax-2eax+3eax+Caaaaa2x21=eaxx2-2+3+C.aaa1113.xsin2xdx=-xdcos2x=-xcos2x+cos2xdx22211=-xcos2x+sin2x+C.24xdx 4.arcsinxdx=xarcsinx-xdarcsinx=xar
12、csinx-1-x21d(1-x2)=xarcsinx+=xarcsinx+1-x2+C.21-x25.arctanxdx=xarctanx-xdarctanx=xarctanx-xdx1+x21d(1+x2)12=xarctanx-=xarctanx-ln(1+x)+C.221+x21116.I=e2xcos3xdx=cos3xde2x=e2xcos3x-e2xdcos3x2221313=e2xcos3x+e2xsin3xdx=e2xcos3x+sin3xde2x222413=e2xcos3x+e2xsin3x-3e2xcos3xdx24139=e2xcos3x+e2xsin3x-I,244
13、4131I=cos3x+sin3xe2x+C=(2cos3x+3sin3x)e2x+C.132413sin3x7.I=xdx=-sin3xde-x=-e-xsin3x+3e-xcos3xdxe()=-e-xsin3x-3cos3xde-x=-e-xsin3x-3e-xcos3x+3e-xsin3xdx ()=-e-xsin3x-3(e-xcos3x+3I), 1e-x-x-xI=(-esin3x-3ecos3x)+C=-(sin3x+3cos3x)+C.10108.I=eaxsinbxdx=11axbaxaxsinbxde=esinbx-ecosbxdxaaa1axbesinbx-2cosbx
14、deaxaa1b=eaxsinbx-2eaxcosbx+beaxsinbxdxaa1b=eaxsinbx-2(eaxcosbx+bI).aa11axbaxI=esinbx-ecosbx,22baa1+2aeaxI=2(asinbx-bcosbx)+C.2a+b()9.I=1+9x2dx=x1+9x2-xd1+9x2=x1+9x2-x18xdx21+9x2dx=x1+9x2-1+9x2dx-21+9xdx=x1+9x2-I-,21+9x111I=x1+9x2+ln(3x+1+9x2)+C22311=x1+9x2+ln(3x+1+9x2)+C.2610.xcoshxdx=xdsinhx=xsinh
15、x-sinhxdx=xsinhx-coshx+C.11.ln(x+1+x2)dx=xln(x+1+x2)-xdln(x+1+x2)=xln(x+1+x2)-xdx1+x2=xln(x+1+x2)-1+x2+C. xarccosx1-x212.(arccosx)2dx=x(arccosx)2+2=x(arccosx)2-2arccosxd1-x2=x(arccosx)2-2dx(1-x2arccosx+1dx)=x(arccosx)2-21-x2arccosx-2x+C. 13.xarccosxdx11=arccosxd(1-x2)221-x2arccosx1dx=+2(1-x2)2(1-x2)
16、1-x2=arccosx1x+C.222(1-x)21-xxdx2(1+x)x14.arctanxdx=xarctanx-=xarctanx-1xdx2.x=u,x=u,dx=2udu 21+xxdxu2udu=1+x1+u2=2(u-arctanu)+C,1arctanxdx=xarctanx-22(x-arctanx)+C=xarctanx-(x-arctanx)+C=(x+1)arctanx-x+C.15.arcsinxarcsinxdx1dx=-arcsinxd=-+x1-x2x2xxarcsinxdx=-+(x0)22xx1/x-1=-arcsinxd(1/x)arcsinx-=-l
17、n|1/x+1/x2-1|+Cxx1/x2-1arcsinx=-+ln(1-1-x2)-lnx+Cxarcsinx=-+ln(1-1-x2)-ln|x|+C(x0)(原函数为偶函数).x1x4(lnx)21x42lnxdx322416.x(lnx)dx=(lnx)dx=-444xx4(lnx)213x4(lnx)21=-xlnxdx=-lnxdx44248x4(lnx)2x413x4(lnx)2x41=-lnx+xdx=-lnx+x4+C.482488xarctanxdx1arctanxd(1+x2)1217.=-arctanxd(1+x2)-3/225/225/2(1+x)2(1+x)23=
18、-arctanx1dx2+.x=tanu,u(-p/2,p/2).dx=secudu, 23/225/23(1+x)3(1+x)dx32=cosudu=(1-sinu)dsinu=(1+x2)5/21x1x=sinu-sin3u+C=-+C,22331+x1+x3xarctanxdxarctanx1x1x-+C (1+x2)5/2=-3(1+x2)3/2+31+x231+x2arctanx1x1x3=-+-+C.3(1+x2)3/231+x29(1+x2)3/2318.xln(x+1+x2)dx=122ln(x+1+x)dx2121x2dx2=xln(x+1+x)-221+x2121(x2+1
19、)-1dx2=xln(x+1+x)-221+x2111dx=x2ln(x+1+x2)-1+x2dx+2221+x2121x1+x2ln(x+1+x2)12=xln(x+1+x)-+ln(x+1+x2)+C22222=1211xln(x+1+x2)-x1+x2+ln(x+1+x2)+C.244习题3.3 求下列不定积分:1.x-1x-1dx=(x+2)(x+4)dx,x2+6x+8x-1AB=+,(x+2)(x+4)x+2x+4 -2-13-4-15A=-,B=,-2+42-4+22x-1-3/25/2=+,(x+2)(x+4)x+2x+4x-135dx=-ln|x+2|+ln|x+4|+C.x
20、2+6x+8223x4+x2+12.I=2dx.x+x-63x4+x2+1-40x+1332=3x-3x+22+,x2+x-6x2+x-6-40x+133-40x+133AB =+,x2+x-6(x+3)(x-2)x+3x-2-40(-3)+133253-402+13353A=-,B=.-3-252+353x2253533I=x-+22x-ln|x+3|+ln|x-2|+C.2552x2-53.I=4dxx-5x2+62x2-52u-52=(u=x)422x-5x+6u-5u+62u-5AB=+,(u-2)(u-3)u-2u-322-523-5A=1,B=1.2-33-22x2-511=+,2
21、24222x-5x+6x-2x-3I=122lnx-21x-3+ln+C.x+223x+34.I=dx.(x-1)2(x-2)1111=-(x-1)2(x-2)x-2x-2x-1=111-, (x-2)2x-2x-11x-1+ln+C. x-2x-2I=-x25.I=dx.41-xx2x21(1+x2)-(1-x2)=1-x4(1-x2)(1+x2)2(1-x2)(1+x2)111=-,2221-x1+x11+x1I=ln-arctanx+C.41-x2dx6.I=3.x+111ABx+C=+,x3+1(x+1)(x2-x+1)x+1x2-x+111A=2=,1+1+13x2-x+11111=
22、+(x+1)(Bx+C)=(B+)x2+(B+C-)x+C+,33331211C+=1,C=,B+=0,B=-.333311-x+21x-2=+=-x3+13(x+1)3(x2-x+1)3(x+1)3(x2-x+1)12x-411(2x-1)-3=-=-.3(x+1)6(x2-x+1)3(x+1)6(x2-x+1)112x-111=-+,223(x+1)6(x2-x+1)213x-+ 221112x-1I=ln|x+1|-ln(x2-x+1)+arctan+C.36337.I=dx111.=1+x41+x4(1+2x2+x4)-2x2(x2+1)2-2x21Ax+BCx+D=2=+,222(x
23、+2x+1)(x-2x+1)x+2x+1x-2x+11=(Ax+B)(x2-2x+1)+(Cx+D)(x2+2x+1),1=(A+C)x3+(B-2A+D+2C)x2+(A-2B+C+2D)x+B+D.A+C=0B-2A+D+2C=0,A-2B+C+2D=0,B+D=1.A=122,B=12,C=-122,D=12.1122x+12-1x+121+x4=x2+2x+1+22x2-2x+1=1x+222+-x+2x2+2x+1x2-2x+1=12x+42222x-22x2+2x+1-x2-2x+1=142(2x+2)+2(2x-2)-2x2+2x+1-x2-2x+1=1(2x+2)(2x-2)4
24、2x2+2x+1-x2-2x+1+1142(x+12)2+12+114(x-112.22)+2I=142lnx2+2x+12x2-2x+1+4(arctan(2x+1)+arctan(2x-1)+C.8.I=x3+x2+2(x2+2)2dx.x3+x2+2x(x2+2)x2-2x+2(x2+2)2=(x2+2)2+(x2+2)2=x12x (x2+2)+(x2+2)-(x2+2)2.I=12ln(x2+2)+1x12arctan2+x2+2+C. mexdxdexdu9.2x=x2xx2e+3e+2e+3e+2u+3u+2 xdu1u+1e+11=-du=ln+C=ln+C.x(u+1)(u+
25、2)u+1u+2u+2e+210.cosxdxdsinxdu=(u=sinx)=222sinx+sinx-6sinx+sinx-6u+u-6du111u-2sinx-2-+C=ln+C.du=ln=(u+3)(u-2)5u-2u+3u+3x3dx1x2dx211.1udux4+x2+2=2x4+x2+2=2u2+u+2=12udu1(2u+1)4-1u2+u+2=4u2+u+2du=1d(u2=4+u+2)11u2+u+2du-4du127 u+2+4=112u4ln(u2+u+2)-27arctan+17+C=112x24+14ln(x+x2+2)-27arctan7+C.12.I=dx(x
26、+2)(x2-2x+2).1ABx+C(x+2)(x2-2x+2)=x+2+x2-2x+2A=11(-2)2-2(-2)+2=10.1(x+2)(x2-2x+2)-110(x+2)=Bx+Cx2-2x+210-(x2-2x+2)Bx+C10(x+2)(x2-2x+2)=x2-2x+2-(x2-2x-8)Bx+C10(x+2)(x2-2x+2)=x2-2x+2-(x+2)(x-4)10(x+2)(x2-2x+2)=Bx+Cx2-2x+2-(x-4)Bx+C110(x2-2x+2)=x2-2x+2,B=-10,C=25.I=110ln|x+2|-1x-4 10x2-2x+2dxsinx+3112x
27、-8ln|x+2|-2dx1010x-2x+211(2x-2)-6=ln|x+2|-2dx1020x-2x+2113dx=ln|x+2|-ln(x2-2x+2)+102010(x-1)2+1113=ln|x+2|-ln(x2-2x+2)+arctan(x-1)+C102010=xdxx2du2=2u.13.I=.tan=u,x=2arctanu,dx=,sinx=22+sinx21+u21+u2x1+tan22du2111+uI=2du=du22 2uu+u+1132+u+1+u222x2tan+122u+122=arctan+C=arctan+C.33332tandxx2du.tan=u,x
28、=2arctanu,dx=,1+sinx+cosx21+u22u1-u2sinx=,cosx=.221+u1+u2du 2111+uI=2du=1+u2+2u+1-u2u+1du2u1-u21+1+u21+u2x=ln|u+1|+C=ln|tan+1|+C.214.I=15.cot4xdx=cot2x(csc2x-1)dx=cot2xcsc2xdx-cot2xdx=-cot2xdcotx-(csc2x-1)dx1=-cot3x+cotx+x+C.3116.sec4xdx=(1+tan2x)dtanx=tanx+tan3x+C. 317.I=cosxdx1-3cosxdx1(-3cosx+5)-
29、5dx=-=-5-3cosx35-3cosx35-3cosxx5dx=-+.335-3cosxx2du1-u2tan=u,dx=,cosx=,2221+u1+u2du 2x5x52du1+uI=-+=-+3(1-u2)33335(1+u2)-3(1-u2)5-1+u2x52dux5dux51d2u=-+2=-+2=-+338u)+2334u+13324u2+1x5x5x=-+arctan2u+C=-+arctan2tan+C.36362cos3xdxcos2xdxdx18.I=.2sinx+cosx1+tanx(1+tanx)(1+tanx)dutanx=u,x=arctanu,dx=,1+u
30、2du2du1+uI=,(1+u)(1+u2)(1+u)(1+u2)2111-u1=+(1+u)(1+u2)22(1+u2)1+u1+u2111-u1-u=+,41+u1+u22(1+u2)2111111uI=ln|1+tanx|+arctanu-ln(1+u2)+arctanu+C224484(1+u)222(1+u)1x111=ln|1+tanx|+ln|cosu|+cos2x+tanxcos2x+C.4244419.sin5xcos2xdx=-sin4xcos2xdcosx=-(1-u2)2u2du121=-(u2-2u4+u6)dx=-u3+u5-u7+C357121=-(cosx)3
31、+(cosx)5-(cosx)7+C.3571-cos2x20.sin6xdx=dx2 1=(1-3cos2x+3cos22x-cos32x)dx83x331-sin2x+(1+cos4x)dx-(1-sin22x)dsin2x8161616x33111=-sin2x+x+sin4x-sin2x-sin32x+C 816164316=+C.=11sin3x+sinx21.sinxcosxdx=sin22xcos2xdx=dx4421=(sin23x+sin2x+2sin3xsinx)dx1611-cos6x1-cos2x=+cos2x-cos4xdx16221111=x+sin2x-sin4x
32、-sin6x+C.1644122421-cos2x1+cos2x另解：sinxcosxdx=dx221=(1+cos22x+2cos2x)(1-cos2x)dx81=(1+cos22x+2cos2x-cos2x-cos32x-2cos22x)dx81 =(1+cos2x-cos22x-cos32x)dx81111=x+sin2x-(1+cos4x)dx-(1-sin22x)dsin2x821616242111111=x+sin2x-x+sin4x-sin2x-sin32x+C82431616111=x-sin4x+sin32x+C.16644822.I=dxx2du.tan=u,x=2arct
33、anu,dx=.2sinx+2cosx21+u2du22dududu1+uI=-=-= 2u2-u-122u2(1-u2)-2u2+2u+215+22u-1+u1+u2215u-+122+C=ln2u+5-1+C.=ln5152u-5-1u-22sinxcosx23.2dx= 4sinx+cosx=tanxudtanx=u2(1+u2)+1du(u=tanx)tan2x(1+tan2x)+11du21dv2=2=(v=u)22u(1+u)+12v(1+v)+11dv122v+1=arctan+C22223313v+2212tan2x+1=arctan+C.331dsin2x1dw2另解:I=2
34、=(w=sinx)2222sinx+(1-sinx)2w+(1-w)1=212w-12sin2x-1=arctan+C=arctan+C.2333123(w-)+22dx124.4=-(1+cot2x)dcotx=-cotx-cot3x+C. sinx3dw1-x1-xdx=dx=arcsinx+1-x2+C. 1+x1-x225.1-x-1665dx.x-1=u,x=1+u,dx=6udu,31+x-1(1-u3)u5duu5-u8-u+16432I=6=6du=-6(u-u-u+u+u+1+)dx 1+u21+u21+u2111111=-6u7-u5-u4+u3+u2+u-ln(1+u2)
35、+arctanu+C.54322726.I=27.x+1+x-1dx=x+1-x-1(x+1+x-1x+1-x-1)()2x+1+x-1)dx2x+2x2-1111=dx=x2+xx2-1-ln(x+x2-1)+C.222228.I=dx3(x+1)2(x-1)43=dx(x2-1)3x-1x+1.3x-1x-1=u,=u3,x+1x+11+u326u2dux-1=(x+1)u,x=-1+,dx=,1-u31-u3(1-u3)2I=6u2du(1-u3)21+u32-1u1-u3xdx=6u3133x+1du=-+C=-+C. 2(2u3)2u2x-112xdx12x-1+1dx=222x-x
36、+32x-x+32x-x+31d(x2-x+3)1dx =+=2222x-x+32111x-+2211=x2-x+3+lnx-+x2-x+3+C.2229.30.I=x1/31/22322dx.(1+x)=u,x=(u-1),dx=3(u-1)(2u)du,1/31/2(1+x)(u2-1)3(u2-1)2(u)duI=6=6(u6-3u4+3u2-1)(u4-2u2+1)duu=6(u10-5u8+10u6-10u4+5u2-1)du51051=6u11-u9+u7-2u5+u3-u+C.9731131.I=xdx4x3+1u24u3duu5(u5+u2)-u2 I=3=43dx=4du3u
37、+1u+1u+132u244434=4u-3du=u3-ln(u3+1)+C=4x-ln(4x+1)+C.u+133332x+3x+x2.4x=u,x=u4,dx=4u3du.32.dx=(2x+1)+2x+x122dx=21x+x2d(x2+x)+21x+x2dx=2x2+x+2dx11x+-221=2x2+x+2lnx+x2+x+C.233.=18116+8xdx2284x-4x+54x-4x+5 28x-4+201d(4x-4x+5)5dxdx=dx+824x2-4x+54x2-4x+54x2-4x+5dx=2+x15dx4x2-4x+5+44x2-x+5/415dx=4x2-4x+5+
38、2441x-+1 2151=4x2-4x+5+lnx-+x2-x+5/4+C44215=4x2-4x+5+ln2x-1+4x2-4x+5+C.44=()34.5-2x+x2dx=22+(x-1)2dx (x-1)22=5-2x+x+2ln(5-2x+x)+C.2习题3.4 求下列各定积分:1xdx111.I=.5-4x=u,-13,11.5-4x=u2,x=(5-u2),dx=-udu,-1425-4x123(5-u)131111123I=4-udu=(5-u)dx=5u-u=.31u288361ln2ln2ln2ln2ln2-xln212.xe-xdx=-xde-x=-xe-x+e-xdx=
39、-e=(1-ln2).00000223.x21-x2dx=01p/20sin2tcos2tdt(x=sint)=p/20131ppsin2t(1-sin2t)dt=I2-I4=-=.242216pppp004.xsinxdx=-xdcosx=-xcosx0+cosxdx=p+sinx0=p.0p5.4012099x2x2+9dx=x+9+ln(x+x2+9)=10+ln3.2220pp1111p3666=dx=sintdt=(1-cos2t)dt=(t-sin2t)0-.2002222641-x46.7.x2p210x3px4-x2dx=4-x2+2arcsin=+.2223091413319
40、334532228.x1-xdx=1-xdx=1-udu=-(1-u)3=-.020208803ppp9.2-p2cosx-cosxdx=2p20320cosx-cosxdx=22cosxsinxdx03p244cosxdcosx=-cosx=.330n32=-2pp11p1p/2p10.2cos2xdx=2cosn2xd2x=cosnudu=cosn(t+)dt020202-p/220,n=2k-1;(-1)np/2n=sin(t)dt=(n-1)!pp/2n2-p/2sin(t)dt=.0n!2n!pn(n+1)! n是偶数;a11.(a2-x2)2dx(x=asint)=2cosn+1tdt= 00n!p n是奇数.(n+1)!212.p/20sin11xdx=610!156=.11!693p/2x53p5p13.sindx=2sin6udu=2=.002642216p1p1131p14.(xsinx)dx=x2(1-cos2x)dx=x-x2dsin2x02023040p2p121pp=-xsin2x0+xsin2xdx64201pp311p=-xdcos2x=-xcos2x+cos2xdx640644001p3pp=-+sin2x0=-.6486415.=0p3p3pp3pp/40tan4xdx=p/40
