Integración directa

<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>  |</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> Integración directa</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> <B> <A NAME="arriba"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>D</B><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><I>e cada regla de derivación se puede deducir una regla correspondiente de integración</I>. La integración directa es aplicable cuando identificamos la función primitiva de forma inmediata; esto es, cuando conocemos la regla de derivación que al aplicarla nos permite hallar el integrando a partir de la función primitiva.</FONT></div> <div> Ejemplo: </div> <div> <IMG SRC="213f0c1d0.gif" border=0 width="268" height="29" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div>  <object width="480" height="385"><param name="movie" value="http://www.youtube.com/p/288D05509B9148B9?hl=es_ES&fs=1"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/p/288D05509B9148B9?hl=es_ES&fs=1" type="application/x-shockwave-flash" width="480" height="385" allowscriptaccess="always" allowfullscreen="true"></embed></object>  </div> <bR> <div> <TABLE BORDER="2" CELLPADDING="2" CELLSPACING="1" WIDTH="595" BORDERCOLORLIGHT="#009900" BORDERCOLORDARK="#33FFCC" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#00CC99 HEIGHT= 39 WIDTH="579"><div>  <A NAME="propiedades_fundamentales_de_la"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#FF0099" FACE="Times New Roman" ><B>Propiedades fundamentales de la antidiferenciación</B></FONT></div> <bR> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 55 ><NOBR><div>  <A NAME="p1_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="21403c1d0.gif" border=0 width="316" height="29" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> Esta propiedad indica que podemos sacar un factor <I><B>constante</B></I> de la integral.</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 63 ><NOBR><div>  <A NAME="p2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="214190380.gif" border=0 width="400" height="56" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 87 ><NOBR><div>  <A NAME="p3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="214237500.gif" border=0 width="567" height="80" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 60 ><NOBR><div>  <A NAME="p4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2143eb350.gif" border=0 width="235" height="53" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="271"><div>  <A NAME="p5a"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2144691d0.gif" border=0 width="105" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 68 ><NOBR><div>  <A NAME="p6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2145dd3d0.gif" border=0 width="477" height="61" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 53 ><NOBR><div>  <A NAME="p7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2146d1270.gif" border=0 width="209" height="39" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="271"><div>  <A NAME="p8"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="22710d1d0.gif" border=0 width="269" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 68 WIDTH="305"><div>  <A NAME="p9"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2147b41d0.gif" border=0 width="180" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP><NOBR><div>  <A NAME="p10"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2148ef3d0.gif" border=0 width="239" height="61" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 WIDTH="305"><div>  <A NAME="p11"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2149b41d0.gif" border=0 width="180" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="271"><div>  <A NAME="p12"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="214aa81d0.gif" border=0 width="168" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 WIDTH="305"><div>  <A NAME="p13"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="214bb01d0.gif" border=0 width="176" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP><NOBR><div>  <A NAME="p14"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="214cbb1d0.gif" border=0 width="187" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 ><NOBR><div>  <A NAME="p15"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="214dc91d0.gif" border=0 width="201" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div>  <A NAME="p16"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="214ed41d0.gif" border=0 width="212" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 52 ><NOBR><div>  <A NAME="p17"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="214fc82d0.gif" border=0 width="200" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div>  <A NAME="p18"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2150bd280.gif" border=0 width="189" height="40" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 52 ><NOBR><div>  <A NAME="p19"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2151d12d0.gif" border=0 width="209" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div>  <A NAME="p20"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2152b81d0.gif" border=0 width="184" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 WIDTH="305"><div>  <A NAME="p21"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2153b81d0.gif" border=0 width="184" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP><NOBR><div>  <A NAME="p22"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2154c01d0.gif" border=0 width="192" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 ><NOBR><div>  <A NAME="p23"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2155c91d0.gif" border=0 width="201" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div>  <A NAME="p24"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2156eb1d0.gif" border=0 width="235" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 ><NOBR><div>  <A NAME="25"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="2157eb1d0.gif" border=0 width="235" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="271"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="36" ALIGN="bottom" WIDTH="271" HSPACE="0" VSPACE="0"></tD> </tR> </TABLE></div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="612" BORDERCOLORLIGHT="#00FFFF" BORDERCOLORDARK="#0000FF" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=3 BGCOLOR=#FFFFCC HEIGHT= 58 WIDTH="600"><div>  <A NAME="ejercicios_resueltos"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>Ejercicios resueltos</B></FONT></div> <div> <B>          </B><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>E</B></FONT><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >fectúe las operaciones de antidiferenciación que se indican, aplicando las propiedades correspondientes en cada caso:</FONT></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 50 WIDTH="198"><div> <A HREF="#1__" TITLE="Integración directa"><U><IMG SRC="2158601d0.gif" border=0 width="96" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#2__" TITLE="Integración directa"><U><IMG SRC="2159a71d0.gif" border=0 width="167" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#3__" TITLE="Integración directa"><U><IMG SRC="215a912b0.gif" border=0 width="145" height="43" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 47 ><NOBR><div> <A HREF="#4__" TITLE="Integración directa"><U><IMG SRC="215b9b1d0.gif" border=0 width="155" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#5__" TITLE="Integración directa"><U><IMG SRC="215c6d1d0.gif" border=0 width="109" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#6__" TITLE="Integración directa"><U><IMG SRC="215d7b280.gif" border=0 width="123" height="40" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 50 WIDTH="198"><div> <A HREF="#7__" TITLE="Integración directa"><U><IMG SRC="215e80270.gif" border=0 width="128" height="39" 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="Integración directa"><U><IMG SRC="215f4d2b0.gif" border=0 width="77" height="43" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><div> <A HREF="#9" TITLE="Integración directa"><U><IMG SRC="216060280.gif" border=0 width="96" height="40" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 42 WIDTH="198"><div> <A HREF="#10" TITLE="Integración directa"><U><IMG SRC="216150230.gif" border=0 width="80" height="35" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="198"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="42" ALIGN="bottom" WIDTH="198" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="198"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="42" ALIGN="bottom" WIDTH="198" HSPACE="0" VSPACE="0"></tD> </tR> </TABLE></div> <bR> <div> <B>S o l u c i o n e s</B></div> <bR> <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="Times New Roman" ><B>1.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT><U>Solución - </U><FONT SIZE="3" COLOR="#009900" FACE="Times New Roman" ><U>Juan Beltrán</U></FONT>:</div> <div> <IMG SRC="21628f6d0.gif" border=0 width="399" height="109" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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="Times New Roman" ><B>2.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>  </B></FONT><U>Solución - </U><FONT SIZE="3" COLOR="#009900" FACE="Times New Roman" ><U>Juan Beltrán</U></FONT>:</div> <div> <IMG SRC="2163545d0.gif" border=0 width="596" height="93" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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="Times New Roman" ><B>3.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>  </B></FONT><U>Solución - </U><FONT SIZE="3" COLOR="#009900" FACE="Times New Roman" ><U>Juan Beltrán</U></FONT>:</div> <div> <IMG SRC="216475c50.gif" border=0 width="629" height="197" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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="Times New Roman" ><B>4.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>  </B></FONT><U>Solución - </U><FONT SIZE="3" COLOR="#009900" FACE="Times New Roman" ><U>Juan Beltrán</U></FONT>:</div> <div> <IMG SRC="2165b3a80.gif" border=0 width="435" height="168" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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="Times New Roman" ><B>5.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>  </B></FONT><U>Solución - </U><FONT SIZE="3" COLOR="#009900" FACE="Times New Roman" ><U>Juan Beltrán</U></FONT>:</div> <div> <IMG SRC="2166bbe40.gif" border=0 width="443" height="228" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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="Times New Roman" ><B>6.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>  </B></FONT><U>Solución - </U><FONT SIZE="3" COLOR="#009900" FACE="Times New Roman" ><U>Juan Beltrán</U></FONT>:</div> <div> <IMG SRC="216715d30.gif" border=0 width="533" height="211" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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="Times New Roman" ><B>7.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B>  </B></FONT><U>Solución - </U><FONT SIZE="3" COLOR="#009900" FACE="Times New Roman" ><U>Juan Beltrán</U></FONT>::</div> <div> <IMG SRC="21682fcc0.gif" border=0 width="559" height="204" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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="#00CC00" FACE="Times New Roman" ><B><IMG SRC="21694d2b0.gif" border=0 width="77" height="43" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="216aac700.gif" border=0 width="428" height="112" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="9"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B><IMG SRC="216b60280.gif" border=0 width="96" height="40" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></FONT></div> <div> <B><IMG SRC="216c61690.gif" border=0 width="353" height="105" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div> <B> <A NAME="10"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B><IMG SRC="216d50230.gif" border=0 width="80" height="35" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B><IMG SRC="216ee7430.gif" border=0 width="487" height="67" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <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>