分部积分法的推广公式

246Vol.24No.6200412AgricultureTechnologyDec.2004202∫∫′-=′dxuvuvdxvu∫∫-=vduuvudvudvdxv=′dvdxv=′v∫vdu∫udv∫++xdxxx2cos)52(3∫+-++xdxxxxx2sin)23(212sin)52(2123∫∫-++++xdxxxdxxxxx2cos232cos)23(412sin)52(2123Cxxxxxxxx+--++++2cos832sin432cos)23(412sin)52(2123uxx=++523dxvxdx′=2cosxv2sin21=uv)(xuu=)(xvv=I)(xu1+n)(xvvdxuvuvuvuuvdxuvnnnnnnn∫∫++---+-++′′′-′′+′-=)1(1)3()2()1()()1()1(L2030=n∫∫′-=′vdxuuvdxvun1∫∫′-=+dxvuuvdxuvnnn)()()1(∫∫--′′-′=′dxvuvudxvunnn)1()1()(∫∫---′′′-′′=′′dxvuvudxvunnn)2()2()1(M∫∫+-=′vdxuvudxvunnn)1()()(∫+vdxun)1(1)1(+-n∫+-+xdxxxxcos)523(2552325+-+xxx2654-+xx6203+x260xx1201200xcosxsinxcos-xsin-xcosxsinxcos-52325+-+xxxxsin2654-+xx-xcosxxsin)620(3+xxcos602-Cxxx+++cos120sin120∫+dxbaxekx)sin()sin(bax+)cos(baxa+)sin(2baxa+-uuu′u′′u′′′)1(+nu)1(+nv)1(+nv)(nv)1(-nv)2(-nvvb204kxekxek1kxek21∫+dxbaxekx)sin(dxbaxekabaxekabaxekkxkxkx∫+-+-+=)sin()cos()sin(12222Cekbaxabaxkdxbaxekkakxkx+⋅+-+=++∫2222)cos()sin()sin(Ceakbaxabaxkdxbaxekxkx+⋅++-+=+∫22)cos()sin()sin(∫dxxx2)(arcsinxlnxarcsinxarccosxarctanux=arcsinuxsin=ududxcos=∫∫=⋅=uduuuduuu2sin21cossin222uu220u2sinu2cos21-u2sin41-u2cos81Cuuuuu+++-)2cos412sin212cos21(212Cuuuuuuuu+-++--=)sin(cos81)cossin2(41)sin(cos4122222Cxxxxxx+-+-+-⋅-=)21(81arcsin121)21()(arcsin412222∫dxxx43)(lnux=lnuex=duedxu=∫∫=⋅=dueudueueuuu44434u34u212uu24240209ue4ue441ue4161ue4641ue42561ue410241Ceuuuuu++-+-=4234)12833231634141(Cxxxxx++-+-=4234)323ln83ln43ln(ln41