北大版高等数学第三章 积分的计算及应用答案 习题3.2

习题3.2

求下列不定积分：21.?xlnxdx?122?lnxdx2?x2lnx?12?xdlnx1x22?x221xx22lnx?12?x?xdx?2lnx?2?xdx?2lnx?4?C.2.?x2eaxdx?1x2deax?12ax1ax212axa?axe?a?edx?axe?2axa?xedx?1?2?xdeax?1x2eax?2xaxaxax2eaxa2aa2e?2a2?edx?1axax2e?2?xdeax?1a2ax2eax?2xeax2xa2?a3ea?C?eax?12?x2?2x??aa2?a3??C.?3.?xsin2xdx??12?xdcos2x??12xcos2x?12?cos2xdx??12xcos2x?14sin2x?C.4.?arcsinxdx?xarcsinx??xdarcsinx?xarcsinx??xdx

1?x2?xarcsinx?12?d(1?x2)?1?x2?C.1?x2?xarcsinx5.?arctanxdx?xarctanx??xdarctanx?xarctanx??xdx1?x2?xarctanx?12?d(1?x2)121?x2?xarctanx?2ln(1?x)?C.6.I??e2xcos3xdx?1?cos3xde2x?12xx22ecos3x?12?e2dcos3x?12x2ecos3x?3?e2xsin3xdx?132x22e2xcos3x?4?sin3xde?12x2x2ecos3x?34?esin3x?3?e2xcos3xdx??12x32ecos3x?e2x94sin3x?4I,I?4?1313?cos3x?sin3x?e2x?C?1?2cos3x?3sin3x2x?24??e?C.?137.I??sin3x?xexdx???sin3xde??e?xsin3x?3?e?xcos3xdx??e?xsin3x?3?cos3xde?x??e?xsin3x?3?e?xcos3x?3?e?xsin3xdx?

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??e?xsin3x?3(e?xcos3x?3I),?x

I?1?x3x?3e?x10??esincos3x??C??e10(sin3x?3cos3x)?C.8.I??eaxsinbxdx?11xbaxa?sinbxdeax?aeasinbx?a?ecosbxdx?1axsinbx?b?cosbxdeaxaea2?1axbaxaesinbx?a2?ecosbx?b?eaxsinbxdx??1axbaesinbx?a?eaxx?bI?.

2cosbI?1?1b2?eaxsinbx?eaxcosbx??aa2?,1?b?a2axI?ea2?b2(asinbx?bcosbx)?C.9.I??1?9x2dx?x1?9x2??xd1?9x2?x1?9x2??x?18xdx21?9x2?x1?9x2???1?9x2dx????dx?1?9x2??

?x1?9x2???I???dx?1?9x2?,?I?112x1?9x2?2?13ln(3x?1?9x2)?C?1x1?9x22?126ln(3x?1?9x)?C.10.?xcoshxdx??xdsinhx?xsinhx??sinhxdx?xsinhx?coshx?C.11.?ln(x?1?x2)dx?xln(x?1?x2)??xdln(x?1?x2)?xln(x?1?x2)??xdx?xln(x?1?x2)?1?x2?C.

1?x212.?(arccosx)2dx?x(arccosx)2?2?xarccosxx1?x2d?x(arccosx)2?2?arccosxd1?x2?x(arccosx)2?2?1?x2arccosx??1dx?2

?x(arccosx)2?21?x2arccosx?2x?C.

13.?xarccosxdx11(1?x2)2?2?arccosxd1?x2?arccosx?1dx2(1?x2)2?(1?x2)1?x2?arccosx2(1?x)?1x22?C.1?x214.?arctanxdx?xarctanx??xdx2(1?x)x?xarctanx?12?xdx?u,x?u21?x.x,dx?2udu

?xdx1?x??u2udu1?u2?2?u?arctanu??C,?arctanxdx?xarctanx?12?2(x?arctanx)?C?xarctanx?(x?arctanx)?C?(x?1)arctanx?x?C.15.?arcsinxx2dx???arcsinxd?1??arcsinx?x?????x?dxx1?x2??arcsinx??dxx(x?0)x21/x2?1??arcsinxarcsinxx??d(1/x)2??x?ln|1/x?1/x2?1|?C1/x?1??arcsinxx?ln(1?1?x2)?lnx?C??arcsinxx?ln(1?1?x2)?ln|x|?C(x?0)(原函数为偶函数).416.?x3(lnx)2dx?12(lnx)214?2lnxdx4?(lnx)dx4?x4?4?xx4242?x(lnx)4?1x3lnxdx?x(lnx)142?4?8?lnxdx44?x(lnx)24?x4lnx?13(lnx)282?xdx?x4?x48lnx?148x?C.217.?xarctanxdx)1?2?2?3(1?x2)5/2?1arctanxd(1?x2?(1?x2)5/2?2?????arctanxd(1?x)/2?3? 3

??arctanx1dx3(1?x2)3/2?3?(1?x2).x?tanu,u?(??/2,?/2).dx?sec25/2udu,

?dx32(1?x2)5/2??cosudu??(1?sinu)dsinu?3?sinu?13xx?3sinu?C??1?1?x23??1?x2??C,?3??xarctanxdxarctanxx(1?x2)5/2??3(1?x2)3/?1?2?3??1?x???1?x23??1?x2????C

???arctanx1x1x33(1?x2)3/2?3?1?x29(1?x2)3/2?C.8.?xln(x?1?x2)dx?12?ln(x?1?x2)dx2?1122x2ln(x?1?x2)?xdx2?1?x2?122x2ln(x?1?x2)?1(x?1)?1dx2?1?x2?12x2ln(x?1?x2)?12x2?1?xdx?12?d1?x2122?2x2ln(x?1?x2)?1?x1?xln(x?1?x)?2??2????1ln(x1?x2)?C?2?2??12x2ln(x?1?x2)?114x1?x2?4ln(x?1?x2)?C.4

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