Integración por sustitución

<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 por sustitución</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> <FONT SIZE="3" COLOR="#00CC00" FACE="Times New Roman" ><B> <A NAME="arriba"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>E</B></FONT>n muchas ocasiones, cuando la integración directa no es tan obvia, es posible resolver la integral simplemente con hacer un cambio de variable adecuado; este procedimiento se conoce como <I>integración por sustitución</I>.</div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="633" BORDERCOLORLIGHT="#33FFCC" BORDERCOLORDARK="#009900" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=3 BGCOLOR=#FFFFCC HEIGHT= 39 WIDTH="621"><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>E</B><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >n los siguientes ejercicios realice la integral que se indica:</FONT></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 WIDTH="205"><div> <A HREF="#1__" TITLE="Integración por sustitución"><U><IMG SRC="21715f1d0.gif" border=0 width="95" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#2__" TITLE="Integración por sustitución"><U><IMG SRC="2172781d0.gif" border=0 width="120" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#3__" TITLE="Integración por sustitución"><U><IMG SRC="2173931d0.gif" border=0 width="147" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 WIDTH="205"><div> <A HREF="#4__" TITLE="Integración por sustitución"><U><IMG SRC="2174631d0.gif" border=0 width="99" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#5__" TITLE="Integración por sustitución"><U><IMG SRC="21756f1d0.gif" border=0 width="111" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#6__" TITLE="Integración por sustitución"><U><IMG SRC="21765b1d0.gif" border=0 width="91" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 51 WIDTH="205"><div> <A HREF="#7__" TITLE="Integración por sustitución"><U><IMG SRC="217769270.gif" border=0 width="105" height="39" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#8__" TITLE="Integración por sustitución"><U><IMG SRC="2178951d0.gif" border=0 width="149" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#9__" TITLE="Integración por sustitución"><U><IMG SRC="2179682c0.gif" border=0 width="104" height="44" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 54 ><NOBR><div> <A HREF="#10__" TITLE="Integración por sustitución"><U><IMG SRC="217a9f1d0.gif" border=0 width="159" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#11__" TITLE="Integración por sustitución"><U><IMG SRC="217b811d0.gif" border=0 width="129" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#12__" TITLE="Integración por sustitución"><U><IMG SRC="217c6d2f0.gif" border=0 width="109" height="47" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 55 WIDTH="205"><div> <A HREF="#13__" TITLE="Integración por sustitución"><U><IMG SRC="217d672d0.gif" border=0 width="103" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#14__" TITLE="Integración por sustitución"><U><IMG SRC="217e64300.gif" border=0 width="100" height="48" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#15__" TITLE="Integración por sustitución"><U><IMG SRC="217f551d0.gif" border=0 width="85" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 58 ><NOBR><div> <A HREF="#16__" TITLE="Integración por sustitución"><U><IMG SRC="2180bd330.gif" border=0 width="189" height="51" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#17__" TITLE="Integración por sustitución"><U><IMG SRC="2181541d0.gif" border=0 width="84" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#18__" TITLE="Integración por sustitución"><U><IMG SRC="2182541d0.gif" border=0 width="84" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 55 WIDTH="205"><div> <A HREF="#19__" TITLE="Integración por sustitución"><U><IMG SRC="2183551d0.gif" border=0 width="85" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#20_" TITLE="Integración por sustitución"><U><IMG SRC="218480300.gif" border=0 width="128" height="48" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#21" TITLE="Integración por sustitución"><U><IMG SRC="2185682c0.gif" border=0 width="104" height="44" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 56 WIDTH="205"><div> <A HREF="#22" TITLE="Integración por sustitución"><U><IMG SRC="218685310.gif" border=0 width="133" height="49" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#23" TITLE="Integración por sustitución"><U><IMG SRC="21877d2d0.gif" border=0 width="125" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#24" TITLE="Integración por sustitución"><U><IMG SRC="2188af2f0.gif" border=0 width="175" height="47" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 47 WIDTH="205"><div> <A HREF="#25" TITLE="Integración por sustitución"><U><IMG SRC="21897d280.gif" border=0 width="125" height="40" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#26" TITLE="Integración por sustitución"><U><IMG SRC="218a67270.gif" border=0 width="103" height="39" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#27" TITLE="Integración por sustitución"><U><IMG SRC="218ba01d0.gif" border=0 width="160" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 44 ><NOBR><div> <A HREF="#28" TITLE="Integración por sustitución"><U><IMG SRC="218cc41d0.gif" border=0 width="196" 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="#29" TITLE="Integración por sustitución"><U><IMG SRC="218dca250.gif" border=0 width="202" height="37" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#30" TITLE="Integración por sustitución"><U><IMG SRC="218e951d0.gif" border=0 width="149" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></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="218f58e10.gif" border=0 width="600" height="225" 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="2190d5c50.gif" border=0 width="469" height="197" 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="21916cd90.gif" border=0 width="620" height="217" 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="2192d3cf0.gif" border=0 width="467" height="207" 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="2193e93f0.gif" border=0 width="489" height="319" 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="2194a3bd0.gif" border=0 width="419" height="189" 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="2195bcc00.gif" border=0 width="444" height="192" 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="#CC0033" FACE="Times New Roman" ><B>8.</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="2196a3b50.gif" border=0 width="419" height="181" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></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="#CC0033" FACE="Times New Roman" ><B>9.</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="2197b5e50.gif" border=0 width="437" height="229" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="10__"><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>10.</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="2198c8b50.gif" border=0 width="456" height="181" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="11__"><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>11.</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="2199a3b50.gif" border=0 width="419" height="181" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="12__"><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>12.</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="219aa3ed0.gif" border=0 width="419" height="237" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="13__"><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>13.</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="219bf4180.gif" border=0 width="500" height="280" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="14__"><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>14.</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="219c2d330.gif" border=0 width="557" height="307" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="15__"><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>15.</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="219dc7b50.gif" border=0 width="455" height="181" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="16__"><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>16.</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="219ea0d50.gif" border=0 width="416" height="213" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="17__"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>17. </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="219fb8f30.gif" border=0 width="440" height="243" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="18__"><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>18.</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="21a000b50.gif" border=0 width="256" height="181" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="19__"><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>19.</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="21a1bbd30.gif" border=0 width="443" height="211" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="20_"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="21a280300.gif" border=0 width="128" height="48" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21a381400.gif" border=0 width="385" height="320" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="21"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="21a4682c0.gif" border=0 width="104" height="44" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21a5fc750.gif" border=0 width="508" height="373" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="22"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="21a685310.gif" border=0 width="133" height="49" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21a719380.gif" border=0 width="537" height="312" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="23"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="21a87d2d0.gif" border=0 width="125" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21a9f8610.gif" border=0 width="504" height="353" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="24"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="21aaaf2f0.gif" border=0 width="175" height="47" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21ab070d0.gif" border=0 width="519" height="269" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="25"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="21ac7d280.gif" border=0 width="125" height="40" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21ad03cd0.gif" border=0 width="259" height="205" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="26"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="21ae67270.gif" border=0 width="103" height="39" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21af9b3c0.gif" border=0 width="667" height="316" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="27"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="21b0881d0.gif" border=0 width="136" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21b1e0dc0.gif" border=0 width="480" height="220" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="28"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="21b2c41d0.gif" border=0 width="196" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21b39be00.gif" border=0 width="411" height="224" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="29"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="21b4ca250.gif" border=0 width="202" height="37" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21b5b8e50.gif" border=0 width="440" height="229" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="30"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="21b6951d0.gif" border=0 width="149" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21b7bc000.gif" border=0 width="444" height="256" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <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>