广义积分反常积分.ppt

返回第四节广义积分(反常积分)一、无穷限的广义积分二、无界函数的广义积分三、小结定理(微积分基本公式)如果)(xF是连续函数)(xf在区间],[ba上的一个原函数,则)()()(aFbFdxxfba???.?1112???dxx?102?????xdx返回定义1设函数)(xf在区间),[??a上连续,取ab?,如果极限????babdxxf)(lim存在,则称此极限为函数)(xf在无穷区间),[??a上的广义积分,记作???adxxf)(.???adxxf)(?????babdxxf)(lim当极限存在时,称广义积分收敛;当极限不存在时,、无穷限的广义积分)(xfy?b问:f(x)在(-∞?b]上的反常积分如何计算?返回类似地,设函数)(xf在区间],(b??上连续,取ba?,如果极限????baadxxf)(lim存在,则称此极限为函数)(xf在无穷区间],(b??上的广义积分,记作???bdxxf)(.???bdxxf)(?????baadxxf)(lim当极限存在时,称广义积分收敛;当极限不存在时,)(xf在区间),(????上连续,如果广义积分???0)(dxxf和???0)(dxxf都收敛,则称上述两广义积分之和为函数)(xf在无穷区间),(????上的广义积分,记作?????dxxf)(.?????dxxf)(????0)(dxxf????0)(dxxf?????0)(limaadxxf?????bbdxxf0)(lim极限存在称广义积分收敛;否则称广义积分发散.?)(????????????xdx解??????21xdx?????021xdx?????021xdx??????0211limaadxx??????bbdxx0211lim??0arctanlimaax??????bbx0arctanlim????aaarctanlim?????bbarctanlim????.22??????????????????dxxx????21sin12dxxx????????????211sinxdx?????????????bbxdx211sinlimbbx???????????21coslim???????????2cos1coslim??返回例3证明广义积分????apxdxe当0?p时收敛,当0?????apxdxe??????bapxbdxelimbapxbpe????????????lim?????????????pepepbpablim????????0,ppeap即当0?p时收敛,当0?>0时,是指数衰减函数pxey??0,??p返回例5证明广义积分???11dxxp当1?p时收敛,当1?,1)1(?p???11dxxp????11dxx?????1lnx,???,1)2(?p???11dxxp???????????111pxp???????1,p因此当1?p时广义积分收敛,其值为11?p;当1?,11??pp返回定义2设函数)(xf在区间],(ba上连续,??,如果极限????badxxf??)(lim0存在,则称此极限为函数)(xf在区间],(ba上的广义积分,记作?badxxf)(.?badxxf)(????badxxf??)(0当极限存在时,称广义积分收敛;当极限不存在时,、无界函数的广义积分)(xfy???a瑕点问:f(x)在[a?b)上的反常积分如何计算????badxxf)(lim??b返回类似地,设函数)(xf在区间),[ba上连续,??,如果极限??????badxxf)(lim0存在,则称此极限为函数)(xf在区间),[ba上的广义积分,记作?badxxf)(???????badxxf)(,称广义积分收敛;当极限不存在时,称广义积分发散.)(xfy???b