北大版高等数学第三章积分的计算及应用答案习题3.3

习题3.322:111.,68(2)(4)1,(2)(4)24213415,,24242213/25/2,(2)(4)24135ln|2|ln|4|.6822xxdxdxxxxxxABxxxxABxxxxxxdxxxCxx求下列不定积分42242222223312..631401333322,664013340133,6(3)(2)3240(3)13325340213353,.32523532535322ln|3|ln|2|.255xxIdxxxxxxxxxxxxxxABxxxxxxABxIxxxxC242224222224222253.562525()565625,(2)(3)232252351,1.23322511,56231213lnln.222233xIdxxxxuuxxxuuuABuuuuABxxxxxxxICxx224..(1)(2)1111(1)(2)221dxIxxxxxxx2111,(2)21xxx11ln.22xICxx24222242222225..11(1)(1)1(1)(1)2(1)(1)111,211111lnarctan.412xIdxxxxxxxxxxxxxxIxCx332222232226..111,1(1)(1)1111,111311111(1)()()(),333312111,,0,.33331121213(1)3(1)3(1)3(1)1243(1)6(dxIxABxCxxxxxxxAxxxBxCBxBCxCCCBBxxxxxxxxxxxxx2222211(21)3.1)3(1)6(1)112111,3(1)6(1)2132211121ln|1|ln(1)arctan.3633xxxxxxxxxxIxxxC44242222222222321117..11(12)2(1)21,(21)(21)21211()(21)()(21),1()(22)(22).0220,220,1dxIxxxxxxxAxBCxDxxxxxxxxAxBxxCxDxxACxBADCxABCDxBDACBADCABCDBD4222222222.1111,,,.222222111112222221212112222212112222224221211(22)2(22)24221211(22)(4221ABCDxxxxxxxxxxxxxxxxxxxxxxxxxxxx222222222)1142111()2211.411()221212lnarctan(21)arctan(21).44221xxxxxxxIxxCxx3222322222222222222228..(2)2(2)22(2)(2)(2)12.(2)(2)(2)111ln(2)arctan.2222xxIdxxxxxxxxxxxxxxxxxIxCx2229.3232321111lnln.(1)(2)1222xxxxxxxxedxdedueeeeuuduueduCCuuuuue222cossin10.(sin)sinsin6sinsin661112sin2lnln.(3)(2)5233sin3xdxdxduuxxxxxuuduuxduCCuuuuux322424222222222421111.22222121(21)142421(2)1142417241121ln(2)arctan42771121ln(2)arctan.4277xdxxdxuduxxxxuuuduuduuuuuduududuuuuuuuCxxxC22222222222212..(2)(22)1(2)(22)22211.(2)2(2)21011(2)(22)10(2)2210(22)10(2)(22)22(28)10(2)(22)22(2)(4dxIxxxABxCxxxxxxABxCxxxxxxxxBxCxxxxxxxBxCxxxxxxx22222)10(2)(22)22(4)12,,.10(22)22105114ln|2|101022BxCxxxxxxBxCBCxxxxxIxdxxx222221128ln|2|10102211(22)6ln|2|102022113ln|2|ln(22)102010(1)1113ln|2|ln(22)arctan(1)102010xxdxxxxxdxxxdxxxxxxxxxC222222222tan22213..tan,2arctan,,sin.2sin2111tan22111211321222tan122122arctanarctan.3333xdxxduuIuxudxxxxuuduuIduduuuuuuxuCC2222222222214..tan,2arctan,,1sincos2121sin,cos.1121112211211111ln|1|ln|tan1|.2dxxduIuxudxxxuuuxxuuduuIduduuuuuuuuuxuCC42222222315.cotcot(csc1)cotcsccotcotcot(csc1)1cotcot.3xdxxxdxxxdxxdxxdxxdxxxxC423116.sec(1tan)tantantan.3xdxxdxxxC2222222222cos13cos1(3cos5)517.53cos353cos353cos5.3353cos21tan,,cos,211255213(1)33335(1)3(1)51525338)2334xdxxdxxdxIxxxxdxxxduuudxxuuduxxduuIuuuuxduxduuu251213324155arctan2arctan2tan.36362xduuxxxuCC322222222222222coscos18..sincos1tan(1tan)(1tan)tan,arctan,,11,(1)(1)(1)(1)1111(1)(1)2(1)111111,4112(1)1xdxxdxdxIxxxxxduxuxudxududuuIuuuuuuuuuuuuuuuI2222211111ln|1tan|arctanln(1)arctan4484(1)222(1)1111ln|1tan|ln|cos|costancos.42444uxuuuCuuxxuxxxC524222224635735719.sincossincoscos(1)121(2)357121(cos)(cos)(cos).357xxdxxxdxuuduuuudxuuuCxxxC36231cos220.sin21(13cos23cos2cos2)8xxdxdxxxxdx23331sin2(1cos4)(1sin2)sin2816161633111sin2sin4sin2sin2816164163.xxxdxxdxxxxxxxCC224222211sin3sin21.sincossin2cos4421sin3sin2sin3sin1611cos61cos2cos2cos416221111sin2sin4sin6.164412xxxxdxxxdxdxxxxxdxxxxxdxxxxxC22422322321cos21cos2sincos221(1cos22cos2)(1cos2)811cos22cos2cos2cos22cos2811cos2cos2cos281111sin2(1cos4)(1sin2)sin2821616xxxxdxdxxxxdxxxxxxdxxxxdxxxxdxxdx另解：33111111sin2sin4sin2sin282164163111sin4sin2.166448xxxxxxCxxxC222222222222..tan,2arctan,.sin2cos2122122(1)222115112215125122lnln.51525122dxxduIuxudxxxududududuuIuuuuuuuuuuuCCuu24sincos23.sincosxxdxxx2222222222222222tantan(tan)tan(1tan)1(1)111()2(1)12(1)111221arctan2233132212tan1arctan.331sin1:(2sin(1sin)2(1)xudxduuxxxuududvvuuuvvdvvCvxCdxdwIwxxww另解2222sin)11212sin1arctanarctan.233313()22xdwwxCCw234124.(1cot)cotcotcot.sin3dxxdxxxCx221125.arcsin1.11xxdxdxxxCxx6563355864322227543221126..1,1,6,11(1)1666(1)1111111116ln(1)arctan.754322xIdxxuxudxuduxuuduuuuIduuuuuudxuuuuuuuuuuuC