Miscelánea 1

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=3 height=24 BGCOLOR="#006666" TEXT="#080000"> <div> <B>C á l c u l o   2 1</B><FONT SIZE="4" COLOR="#FFCC66" FACE="Bell MT" ><B> </B></FONT><FONT SIZE="4" COLOR="#FFFFCC" FACE="Bell MT" >   </FONT><B> </B> |   <A HREF="index.htm" TARGET="_top" TITLE="Cálculo 21"><U>Home</U></A>  |  <A HREF="id98.htm" TARGET="_top" TITLE="Contenido"><U>Contenido</U></A>  |  <A HREF="id341.htm" TARGET="_top" TITLE="Introducción al cálculo integral"><U>Cálculo integral</U></A>  |  <A HREF="id466.htm" TARGET="_top" TITLE="Técnicas de integración"><U>Técnicas de integración</U></A>  |  <A HREF="id477.htm" TARGET="_top" TITLE="Misceláneas"><U>Misceláneas</U></A>  |</div> <div> </div> </td> </tr><tr> <td valign="top" rowspan="2" width=160 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B> <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3073775660"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </B></div> </td> <td valign="top" height=432 colspan=2 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> Miscelánea 1</div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3926688482" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="612" BORDERCOLORLIGHT="#33FFCC" BORDERCOLORDARK="#009900" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=3 HEIGHT= 20 WIDTH="600"><div> <B> <A NAME="enunciados_de_los_ejercicios"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Enunciados de los ejercicios</B></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 52 WIDTH="198"><div> <A HREF="#1" TITLE="Miscelánea 1"><U><IMG SRC="2276481f0.gif" border=0 width="72" height="31" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP COLSPAN=2><NOBR><div> <A HREF="#2_" TITLE="Miscelánea 1"><U><IMG SRC="2277512d0.gif" border=0 width="337" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 54 WIDTH="198"><div> <A HREF="#3__" TITLE="Miscelánea 1"><U><IMG SRC="227861290.gif" border=0 width="97" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#4_" TITLE="Miscelánea 1"><U><IMG SRC="22795c290.gif" border=0 width="92" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#5_" TITLE="Miscelánea 1"><U><IMG SRC="227a8d2f0.gif" border=0 width="141" height="47" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 52 WIDTH="198"><div> <A HREF="#6_" TITLE="Miscelánea 1"><U><IMG SRC="227b5f290.gif" border=0 width="95" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#7" TITLE="Miscelánea 1"><U><IMG SRC="227c67290.gif" border=0 width="103" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#8" TITLE="Miscelánea 1"><U><IMG SRC="227d802d0.gif" border=0 width="128" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 48 WIDTH="198"><div> <A HREF="#9" TITLE="Miscelánea 1"><U><IMG SRC="227e711f0.gif" border=0 width="113" height="31" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#10" TITLE="Miscelánea 1"><U><IMG SRC="227f85290.gif" border=0 width="133" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#11" TITLE="Miscelánea 1"><U><IMG SRC="2280671d0.gif" border=0 width="103" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 48 WIDTH="198"><div> <A HREF="#12" TITLE="Miscelánea 1"><U><IMG SRC="228161230.gif" border=0 width="97" height="35" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#13" TITLE="Miscelánea 1"><U><IMG SRC="22826b230.gif" border=0 width="107" height="35" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#14" TITLE="Miscelánea 1"><U><IMG SRC="22835f290.gif" border=0 width="95" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 48 WIDTH="198"><div> <A HREF="#15" TITLE="Miscelánea 1"><U><IMG SRC="228460230.gif" border=0 width="96" height="35" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#16" TITLE="Miscelánea 1"><U><IMG SRC="228568290.gif" border=0 width="104" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#17" TITLE="Miscelánea 1"><U><IMG SRC="228650290.gif" border=0 width="80" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> </TABLE></div> <div> <B>S o l u c i o n e s</B></div> <div>  <A NAME="1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Arial" ><B>1</B></FONT><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>.  </B></FONT><U>Solución:</U></div> <div> <B><IMG SRC="228781700.gif" border=0 width="385" height="368" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="2_"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Arial" ><B>2</B></FONT><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>.  </B></FONT><U>Solución:</U></div> <div> <U><IMG SRC="2288810b0.gif" border=0 width="641" height="267" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div>  <A NAME="3__"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Arial" ><B>3</B></FONT><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>.  </B></FONT><U>Solución:</U></div> <div> <U><IMG SRC="228983c80.gif" border=0 width="387" height="200" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div>  <A NAME="4_"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Arial" ><B>4</B></FONT><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>.  </B></FONT><U>Solución:</U></div> <div> <U><IMG SRC="228a5cd50.gif" border=0 width="348" height="213" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div>  <A NAME="5_"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Arial" ><B>5</B></FONT><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>.  </B></FONT><U>Solución:</U></div> <div> <U><IMG SRC="228b25230.gif" border=0 width="549" height="547" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div>  <A NAME="6_"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Arial" ><B>6</B></FONT><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>.  </B></FONT><U>Solución:</U></div> <div> <U><IMG SRC="228cbfcb0.gif" border=0 width="447" height="203" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div>  <A NAME="7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Arial" ><B>7</B></FONT><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>.  </B></FONT><U>Solución:</U></div> <div> <U><IMG SRC="228de3f40.gif" border=0 width="483" height="244" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div>  <A NAME="8"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Arial" ><B>8</B></FONT><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>.  </B></FONT><U>Solución:</U></div> <div> <U><IMG SRC="228e2f850.gif" border=0 width="559" height="901" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div>  <A NAME="9"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><U><IMG SRC="228f711f0.gif" border=0 width="113" height="31" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <div> <U><IMG SRC="229077630.gif" border=0 width="631" height="355" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div>  <A NAME="10"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><U><IMG SRC="229185290.gif" border=0 width="133" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <div> <U><IMG SRC="22924dbd0.gif" border=0 width="589" height="957" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div>  <A NAME="11"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><U><IMG SRC="2293671d0.gif" border=0 width="103" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <div> <U><IMG SRC="2294b9070.gif" border=0 width="441" height="263" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div> <U> <A NAME="12"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></U><U><IMG SRC="229561230.gif" border=0 width="97" height="35" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <div> <U><IMG SRC="2296a0f00.gif" border=0 width="416" height="240" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div> <U> <A NAME="13"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></U><U><IMG SRC="22976b230.gif" border=0 width="107" height="35" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <div> <U><IMG SRC="2298fbd90.gif" border=0 width="251" height="217" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div> <U> <A NAME="14"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></U><U><IMG SRC="22995f290.gif" border=0 width="95" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <div> <U><IMG SRC="229a67e00.gif" border=0 width="615" height="224" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div> <U> <A NAME="15"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></U><U><IMG SRC="229b60230.gif" border=0 width="96" height="35" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <div> <U><IMG SRC="229c61f00.gif" border=0 width="609" height="240" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div> <U> <A NAME="16"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></U><U><IMG SRC="229d68290.gif" border=0 width="104" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <div> <U><IMG SRC="229e58f40.gif" border=0 width="344" height="244" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <div> <U> <A NAME="17"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></U><U><IMG SRC="229f50290.gif" border=0 width="80" height="41" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <div> <U><IMG SRC="22a040a80.gif" border=0 width="576" height="936" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <div> <U><IMG SRC="22a1a4d30.gif" border=0 width="676" height="467" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></div> <bR> <bR> <bR> <bR> </td> </tr><tr> <td valign="top" colspan=2 height=24 BGCOLOR="#006666" TEXT="#080000"> <div> <B> <form action="http://www.google.com.co" id="cse-search-box" target="_blank"> <div> <input type="hidden" name="cx" value="partner-pub-6266400581769473:4577786115" /> <input type="hidden" name="ie" value="UTF-8" /> <input type="text" name="q" size="100" /> <input type="submit" name="sa" value="Buscar" /> </div> </form> <script type="text/javascript" src="http://www.google.com.co/coop/cse/brand?form=cse-search-box&lang=es"></script>  </B></div> </td> </tr></table></body>