北大版高等数学第三章 积分的计算及应用答案 习题32.doc

1习题习题 3.22 222222 22222222 2222111.lnlnlnln222 111lnlnln.222224 111122.1212212axaxaxaxaxaxaxaxaxaxaxaxxxxdxxdxxx dxxxxxxxdxxxdxxCxx e dxx dex ee dxx exe dxaaaaa xx exdex eee dxaaaaax eag求下列不定积分：2 2232 23222122122.1113.sin2cos2cos2cos2222 11cos2sin2.244. arcsinarcsinarcsinarcsin 1 1(1)arcsin21axaxaxaxaxxxdex eeeCaaaa xexCaaaxxdxxdxxxxdxxxxCxdxxdxxxxdxxx x dxxxx x      2arcsin1.xxC22 2 2222222225. arctanarctanarctanarctan1 1(1)1arctanarctanln(1).212 1116.cos3cos3cos3cos3222 1313cos3sin3cos3sin32224 1xxxxxxxxxdxxdxxxxdxxxx dxxxxxxCxIexdxxdeexe dxexexdxexxde  22222223cos3sin33cos324 139cos3sin3,244 4131cos3sin32cos33sin3.13 2413 sin37.sin3sin33cos3sin33 cos3sin33xxxxxxxxxx xxxxxexexexdxexexIIxx eCxx eCxIdxxdeexexdxeexxdeexe    cos33sin3xxexdx2sin33(cos33 ),1sin33cos3(sin33cos3 ).1010xxx xxexexIeIexexCxxC   22222222118.sinsinsincos1sincos1sincossin1sincos.11sincos, 1( sincoaxaxaxaxaxaxaxaxaxaxaxaxaxaxbIebxdxbxdeebxebxdxaaa bebxbxdeaa bebxebxb ebxdxaa bebxebxbIaa bIebxebxbaa a eIabxbabs).bxC222222222222229.1 91 91 9181 9 2 1 91 91 9 1 91 9, 1 9 11 11 9ln(31 9)22 3 111 9ln(31 9).26Ix dxxxxdxxxdxxx xdxxxx dx xdxxxI xIxxxxCxxxxC gg2222222222210.coshsinhsinhsinhsinhcosh.11. ln(1)ln(1)ln(1)ln(1)ln(1)1. 1 arccos12. (arccos )(arccos )2 1(arccos )2 arccoxxdxxdxxxxdxxxxCxxdxxxxxdxxxdxxxxxxxxC x xxx dxxxdx xxx  222s1(arccos )21arccos1xdxxxxxdx 322(arccos )2 1arccos2.xxxxxC2222222222arccos1113.arccos(1)21 arccos1 2(1)2(1) 1 arccos1.2(1)2114. arctanarctan2(1)1arctan.,,22122arctan,11arcxxdxxdxx xdx xxx xxCxx xdxxdxxxxxxdxxxxu xudxuduxxdxu uduuuCxu1tanarctan2(arctan)2 arctan(arctan)(1)arctan.xdxxxxxCxxxxCxxxCg22222222arcsin1arcsin15.arcsin 1 arcsin(0) 1/1 arcsin(1/ )arcsinln |1/1/1| 1/1 arcsinln(11)lnarcsinln(11)ln ||(0)(xxdxdxxdxxxxx xdxxxxx xdxxxxCxxx xxxCx xxxC xx        原函数为偶函数424 32244242 34424424 34).1(ln )12ln16.(ln )(ln )444 (ln )1(ln )1lnln4248 (ln )1(ln )1lnln.482488xxxxdxxxdxxdxx xxxxxxdxxdxxxxxxxxx dxxxCg2 23/2 2 5/22 5/2arctan1arctan(1)1217.arctan(1)(1)2(1)23xxdxxdxxdxxxg42 2 3/22 5/2arctan1.tan ,(/2,/2).sec,3(1)3(1)xdxxu udxuduxx  32 2 5/2332232 5/22 3/22232 3/22 3/22cos(1 sin) sin(1)11sinsin,3311arctanarctan11 (1)3(1)3311arctan11.3(1)39 (1)1dxuduu duxxxuuCC xxxxdxxxxCxxxxxxxCxxx  2222 2222 222222222 2222118.ln(1)ln(1)2 11ln(1)221 11(1) 1ln(1)221 111ln(1)12221111ln(1)1ln(1)ln(1)222221ln(12xxxdxxxdxx dxxxx x xdxxxx x dxxxxx dx xxxxxxxxxxCxx    22211)1ln(1).44xxxxxC。