復(fù)旦數(shù)學(xué)分析附件習(xí)題答案與提示

習(xí)題∞⒈做的功為電場(chǎng)在該點(diǎn)處的電位。一個(gè)帶電量q的電場(chǎng)對(duì)距離rFkr2（k為常數(shù)x處的電位。 f(x)dx和 g(x)dx收斂，k1和k2為常數(shù)，xqq⒉g(x)dx也收斂kf(x)12[k1f(xk2g(x)]dxk1 f(x)dxk2習(xí)題∞⒈做的功為電場(chǎng)在該點(diǎn)處的電位。一個(gè)帶電量q的電場(chǎng)對(duì)距離rFkr2（k為常數(shù)x處的電位。 f(x)dx和 g(x)dx收斂，k1和k2為常數(shù)，xqq⒉g(x)dx也收斂kf(x)12[k1f(xk2g(x)]dxk1 f(x)dxk2 g(x)dx⒊002 sin5xdx cos2xdx⑴⑵ (x2a2)(x2b20 dx;⑶⑷x2x(a0,b0) xlnp2x (aR) (pR)⑸⑹02 dx; dx⑺⑻(x21)3/(exex0 ln1dxdx⑼⑽1xx⒋1 e dx;⑴⑵1xx1ln2 1dx;⑶⑷0(2x)1x1 tan112dx⑸;⑹1 x0n求極限。⒌n⒍1200xlnsinxdxlncosxdx1arcsin xcotxdx2dxx00ln 1x10dx⒎1 (cpv)1x2dx x2dx⑴⑵1 ⑶xlnxdx⒏⑴以 f(x200xlnsinxdxlncosxdx1arcsin xcotxdx2dxx00ln 1x10dx⒎1 (cpv)1x2dx x2dx⑴⑵1 ⑶xlnxdx⒏⑴以 f(x)dx為例，敘述并證明反常積分的線性性，保序性和區(qū)間可加性⒐10.a0fxa1dxaln dxln x x0011．設(shè) f(x)dx收斂，且limf(x)A。證明A0)上可導(dǎo)， f(x)dx f(x)dx都收斂，證明limf(x)012f(x在aa在教師的指導(dǎo)下，試編制一個(gè)通用的Gauss-Legendre求積公式程序，在電子計(jì)算機(jī)1ln(1（精確值 dx⑴x621 lnxln(1x)dx（精確值2 60（精確值 ln2 lncosxdx22 20tanxdx⑷0 sin(x2)dx2202習(xí)題⒈⑴證明比較判別法（8.2.2；⑵舉例說(shuō)明當(dāng)比較判別法的極限形式中l(wèi)0或時(shí)，a(x)dx和 f8.2.3⒉⒊arctan1 dxdx⑴⑵1x3e2lnx1dx（p,qRdx1x⑶⑷11x|sin習(xí)題⒈⑴證明比較判別法（8.2.2；⑵舉例說(shuō)明當(dāng)比較判別法的極限形式中l(wèi)0或時(shí)，a(x)dx和 f8.2.3⒉⒊arctan1 dxdx⑴⑵1x3e2lnx1dx（p,qRdx1x⑶⑷11x|sinxfx(cpv)fx)dx收斂與fx)dx收斂是等價(jià)的。：⒋⒌lnlnlnsin2dx（pRsinxdx⑴⑵ xsinxarctanxdx（pR0sin(x)dx2⑶⑷x1pm(x)sin（px和⑸mnqnfx在[a,b]xb⒍設(shè)⒎1lnxdx1 0x2(1x);0x2⑴⑵1cos dxdx⑶2⑷2cos2xsin2x0011(1x dx |lnx|dxxp⑹001 |lnx|dx⑺0⒏1xp110dxdx(p,qR⑴⑵x(x1)2(xln33arctanxdx;⑶⑷x x0tan0/ dxp1dx⑸⑹0x dxdx⑺⑻2 xxplnqxxsinqdx(p0dx⑴⑵1x1xarctanxdx;⑶⑷x x0tan0/ dxp1dx⑸⑹0x dxdx⑺⑻2 xxplnqxxsinqdx(p0dx⑴⑵1x1x01sinsin cos sindxdx⑶⑷00xxsinx1 x111 dx 0x x(p01x4 xsinxsinxdx收斂0111．設(shè)f(x)單調(diào)，且當(dāng)x0時(shí)f(x)，證明 f(x)dx收斂的必要條件0limxf(x)012．設(shè) f(x)dx收斂，且xf(x)在[a,)上單調(diào)減少，證明:limx(lnx)f(x)013fxlimfx)0fx在[0,0fxsin2xdxf(x)0，證 f2(x)dx收斂14. f(x)dx絕對(duì)收斂，且