Integración por sustitución
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<bR> <bR> <div> <A HREF="id30_m.htm#arriba" TARGET="TRLX_Middle" TITLE="Integración por sustitución"><U><!-- Start Gem Animated Image (GIF) --> <img src="HTMLobj-287/aniGif.gif" alt="Arriba" border="0" align="bottom" width="50" height="50"> <!-- End Gem Animated Image (GIF) --> </U></A></div> </td> <td valign="top" height=382 BGCOLOR="#FFFFFF" TEXT="#080000"> <bR> <div> <B>Integración por sustitución</B></div> <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=#00CCCC 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="1205f1d0.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="121781d0.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="122931d0.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="123631d0.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="1246f1d0.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="1255b1d0.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="12669270.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="127951d0.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="128682c0.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="1299f1d0.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="12a811d0.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="12b6d2f0.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="12c672d0.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="12d64300.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="12e551d0.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="12fbd330.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="130541d0.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="131541d0.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="132551d0.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="13380300.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="134682c0.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="13585310.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="1367d2d0.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="#25" TITLE="Integración por sustitución"><U><IMG SRC="137af2f0.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="1387d280.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="13967270.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="13aa01d0.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="13bc41d0.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="345ca250.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="13d951d0.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> Solución:</div> <div> <IMG SRC="13e58e10.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>Solución:</div> <div> <IMG SRC="13fd5c50.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>Solución:</div> <div> <IMG SRC="1406cd90.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>Solución:</div> <div> <IMG SRC="141d3cf0.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>Solución:</div> <div> <IMG SRC="142e93f0.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>Solución:</div> <div> <IMG SRC="143a3bd0.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> Solución:</div> <div> <IMG SRC="144bcc00.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>Solución:</div> <div> <IMG SRC="145a3b50.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>Solución:</div> <div> <IMG SRC="146b5e50.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>Solución:</div> <div> <IMG SRC="147c8b50.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>Solución:</div> <div> <IMG SRC="148a3b50.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>Solución:</div> <div> <IMG SRC="149a3ed0.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>Solución:</div> <div> <IMG SRC="14af4180.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>Solución:</div> <div> <IMG SRC="14b2d330.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>Solución:</div> <div> <IMG SRC="524c7b50.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>Solución:</div> <div> <IMG SRC="14da0d50.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>Solución:</div> <div> <IMG SRC="14eb8f30.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>Solución:</div> <div> <IMG SRC="14f00b50.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>:</div> <div> <IMG SRC="150bbd30.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="15180300.gif" border=0 width="128" height="48" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="15281400.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="153682c0.gif" border=0 width="104" height="44" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="154fc750.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="15585310.gif" border=0 width="133" height="49" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="15619380.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="1577d2d0.gif" border=0 width="125" height="45" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="158f8610.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="159af2f0.gif" border=0 width="175" height="47" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="15a070d0.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="15b7d280.gif" border=0 width="125" height="40" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="15c03cd0.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="15d67270.gif" border=0 width="103" height="39" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="4c19b3c0.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="15f881d0.gif" border=0 width="136" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="160e0dc0.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="161c41d0.gif" border=0 width="196" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="3469be00.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="347ca250.gif" border=0 width="202" height="37" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="348b8e50.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="165951d0.gif" border=0 width="149" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="166bc000.gif" border=0 width="444" height="256" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <bR> <bR> <bR> <bR> </td> <td valign="top" rowspan="2" width=14 BGCOLOR="#009999" TEXT="#080000"> <bR> </td> </tr><tr> <td valign="top" height=21 BGCOLOR="#009999" TEXT="#080000"> <div> <B>| </B><A HREF="id30_m.htm#ejercicios_resueltos" 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