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定积分习题课一、基本概念、基本理论1、定积分的定义2、存在定理(可积的两个充分条件)定理1当函数)(xf在区间],[ba上连续时,称)(xf在区间],[ba上可积.定理2设函数)(xf在区间],[ba上有界,且只有有限个间断点,则)(xf在区间],[ba上可积.3、定积分的性质性质1:badxxgxf)]()([badxxf)(badxxg)(性质2:babadxxfkdxxkf)()((k为常数)性质3:假设bcabadxxf)(bccadxxfdxxf)()(性质4:dxba1dxbaab性质5:如果在区间],[ba上0)(xf,则0)(dxxfba)(ba推论:(1)如果在区间],[ba上)()(xgxf,则dxxfba)(dxxgba)()(ba(2)dxxfba)(dxxfba)()(ba性质6:设M和m分别是)(xf在区间],[ba上的最大值及最小值,则)()()(abMdxxfabmba.性质7:(定积分中值定理)如果函数)(xf在闭区间],[ba上连续,则在积分区间],[ba上至少存在一个点,使dxxfba)())((abf)(ba4、牛顿—莱布尼茨公式5、定积分的计算法(1)换元法:dtttfdxxfba)()]([)((2)分部积分法:bababavduuvudv][6、广义积分(1)无穷限的广义积分adxxf)(babdxxf)(limbdxxf)(baadxxf)(lim当极限存在时,称广义积分收敛;当极限不存在时,称广义积分发散.(2)无界函数的广义积分badxxf)(badxxf)(lim0badxxf)(badxxf)(lim0badxxf)(cadxxf)(bcdxxf)(cadxxf)(lim0bcdxxf)(lim0当极限存在时,称广义积分收敛;当极限不存在时,称广义积分发散.二、典型例题与方法展示求下列极限例12222241241141[).1(limnnnnn))1(sin2sin(sin1).2(limnnnnnn])(41)(41)2(41141[1)1(2222nnninnn原式解:上的,在区间从上式可以看出它是]10[41)(2xxf一个积分和式,于是102102102)2()2(11)2(12141xdxdxxdxx原式21arctan2arctan10x)0)1(sin2sin(sin1)2(limnnnnnn原式)sin)1(sin2sin(sin1limnnnnnnnn上的,在区间从上式可以看出它是]0[sin)(xxf一个积分和式,于是02sin1xdx原式等式不通过计算证明下列不例221212120222)2(21sin)1(2dxeexdxx解:xxxxxsinsinsin]20[)1(2且上,在821sin220220202xxdxdx0]210[0]0,21[)2(yy上,在上在)21(1]2121[212xeex,最小值为的最大值为上,在]2121[1]2121[2121212dxeex于是设)(处处连续,函数若例51)(3x)1()()(21)(02fdtttxxfx试计算证明:再求导分析:先展开提出xdtttxtxxfx)()2(21)(022xxxdtttdtttxdttx02002)()(2)([21)()(2)(2)()(2[21)(22002xxxxdtttxxdttxxfxxxxdtttdttx00)()()()()()(0xxxxdttxfx5)1()1()()(fxxfdxxdxtgxtgxxx0sin000sin4lim求极限例解:xtgxxxtgx200sec)sin(cos)(sinlim原式2100])sin(sinsin)(sin[limtgxtgxtgxxxxtgx1)(21)()(5220xfdttfxfxx求处处连续,且设例xxxfx22ln22)(22解:两边求导2ln2)(2ln2)(22xxxfxf例6.2sin120dxx求解:20cossindxxx原式2440)cos(sin)sin(cosdxxxdxxx.222例7.])1(ln1sin[212128dxxxx求解:dxx2121)1ln(0原式dxxdxx210021)1ln()1ln(.21ln23ln23例8.},1min{222dxxx求解:1,11,},1min{22xxxxxx是偶函数,dxxx},1min{2220原式21102122dxxdxx.2ln232例9.)()1(,)(102022dxxfxdyexfxyy求设解:10022][)1(2dxdyexxyy原式10231002322)1(31])1(31[dxexdyexxxxyy1021)1(2])1[()1(612xdexx=016duueeu).2(61e例10.cos1)(sin2cos1)(sin:,],0[)(0202dxxxfdxxxxfxf证明上连续在设证:,tx令,dtdx)(cos1)(sin)(02dtttft左边dxxxfx02cos1)(sin)(dxxxxfdxxxf0202cos1)(sincos1)(sindxxxfdxxxxf0202cos1)(sincos1)(sin2即.cos1)(sin2cos1)(sin0202dxxxfdxxxxf例11.)()()(.0)(],[)(2abxfdxdxxfxfbaxfbaba证明上连续,在设证:作辅助函数,)()()()(2axtfdtdttfxFxaxa)(2)(1)()(1)()(axxfdttfdttfxfxFxaxa,2)()()()(xaxaxadtdtxftfdttfxf,0)(xf2)()()()(xftftfxf0)2)()()()(()(dtxftftfxfxFxa即.)(单调增加xF,0)(aF又,0)()(aFbF.)()()(2abxfdxdxxfbaba即例12.123)2(;94)1(:2122xxxdxxxdx求下列广义积分解:(1)02029494xxdxxxdx原式bbaaxdxxdx02025)2(lim5)2(limbbaaxx0052arctan51lim52arctan51lim.5(2),1231lim)(lim211xxxxfxx.)(1的瑕点为xfx2120123limxxxdx原式])11(2)11([lim21220xxd210211arcsinlimx.43arcsin2
本文标题:同济高数定积分习题课
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