2022年北大版高等数学第三章积分的计算及应用答案习题.docx

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1、精选学习资料 - - - - - - - - - 习题 3.2 求以下不定积分：名师归纳总结 1.xlnxdx1ln2 xdxx2lnx12 x dlnxxdx第 1 页,共 4 页2222 xlnx1x2g 1xdxx2lnx1xdxx2lnx2 xC.2222242.2 axx e dx12 axx de12 axx e1ax 2e dx12 axx e2ax xe dxaaaaa1x e 2 ax2xde ax1x e 2 ax2 xe ax2e dx axaa2aa2a212 axx e2ax xde12 axx e2 xax e2ax eCaa2aa2a3ax e1x22x2C .a

2、a23 a3.xsin 2 xdx1xdcos2 x1xcos2x1cos2xdx2221xcos2x1sin2xC.244. arcsinxdxxarcsinxxdarcsinxxarcsinxxdx12 xxarcsinx1d1x2xarcsinx12 xC.21x25. arctanxdxxarctanxxdarctanxxarctanx1xdxx2xarctanx1d1x2xarctanx1ln1x2C.21x226. I2 excos3xdx1cos3xde2x12 excos3x12 xe dcos3x22212 excos3x3e2xsin 3xdx12 excos3x3sin

3、32 xdex222412 excos3x3e2xsin 3 x32 excos3xdx2412 excos3x32 exsin 3x9I,244I41cos3x3sin 32 x exC12cos3x3sin 3x e2xC.1324137. Isin 3xdxsin 3xdexexsin 3x3excos3xdxexexsin 3x3 cos3xdexexsin3x3excos3x3exsin 3- - - - - - -精选学习资料 - - - - - - - - - exsin3 x3 excos3x3 ,名师归纳总结 I1exsin 3 x3 excos3 xCexsin 3 x3c

4、os3 C .第 2 页,共 4 页10108.Ieaxsinbxdx1sinax bxde1ax esinbxbax ecosbxdxaaa1ax esinbxbcosax bxdeaa21ax esinbxbeaxcosbxax b esinbxdxa2a1ax esinbxbeaxcosbxbI.aa2I1121ax esinbxbax ecosbx,baa21x22aIa2ax eb2asinbxbcosbxC.9. I192 x dxx19x2xd19x2x19x2xg 18xdx2 19x2x19x2192 x dx1dxx29x19x2I1dxx2,9I1x19x21 1 g2

5、3ln3x19x2C21x19x21ln3x19x2C.2610.xcoshxdxxdsinhxxsinhxsinhxdxxsinhxcoshxC .11. lnx12 x dxxlnx1x2xdlnxxlnx1x2xdx2xlnx12 x1x2C .1x2 12. arccos x dxx arccos 22xarccos xdx1x2x arccos 22 arcco sxd12 xx arccos 2212 xarccos x1 dx- - - - - - -精选学习资料 - - - - - - - - - 2 x arccos 2 1x2arccos x2xC .名师归纳总结 13.x

6、arccosxdx1arccosxd112.23/ 2第 3 页,共 4 页1x222xarccosx1dx21x221x2 12 xarccosx11xx2C.21x2214. arctanxdxxarctanx21xdxxx xarctanx11xdx.xu xu2,dx2 udu2x1xdx xu udu2uarctanuC,1u2arc tanxdxxarctanx1g 2xarctanxC2xarctanxxarctanxCx1arctanxxC.15.arcsinxdxarcsinxd1arcsinxxdxx2x2xx1arcsinxx2dx21x0x1/xarcsinxd1/x

7、1arcsinxln |1/x1/x21 |Cx1/x2xarcsinxln11x2lnxCxarcsinxln11x2ln |x|C x0原函数为偶函数x16.3 xlnx 2dx1lnx2dx4x4lnx21x4g 2lnxdx444xx4lnx21x3lnxdxx4lnx21lnxdx44248x4lnx2x4lnx13 x dxx4lnx 2x4lnx1x4C.482488x17.xarctanxdx1arctanxd1x21g2arctanxd11x25/ 221x25/ 223- - - - - - -精选学习资料 - - - - - - - - - arctanx11dx5/ 2

8、.xtan , u u/ 2,/ 2. dx2 secudu,31x23/ 23x2名师归纳总结 1dx5/ 23 cosudu1sin2u dsinu23C12 xC第 4 页,共 4 页x2sinu1sin3uC1xx211xx23C,33xarctanxdxarctanx1x1x1x25/ 231x23/ 231x231xarctanx11xx21x33/ 2C.31x23/ 239 1x21lnx18.xlnx1x2 dx1lnx1x2dx 2212 xlnx12 x12 x dx2221x12 xlnx12 x1x21 1 dx221x212 xlnx12 x112 x dx1dx22221x12 xlnx12 x1x12 xlnx1x22222212 xlnx12 x1x1x21lnx1x2C.244- - - - - - -