5.1

<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> <FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B>M</B></FONT><FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B>atemáticas de la U</B></FONT><B> (MatU)  | </B><A HREF="http://www.jcbmat.com/" TARGET="_top" TITLE="http://www.jcbmat.com/"><U><B>Home</B></U></A><B> | </B><A HREF="index.htm" TARGET="_top" TITLE="Cálculo 21"><U><B>Cálculo de una variable</B></U></A><B> | </B><A HREF="id543.htm" TARGET="_top" TITLE="D.G. Zill"><U><B>D. G. Zill</B></U></A><B> | </B><A HREF="id546_m.htm#apuntes" TARGET="TRLX_Middle" TITLE="5.1"><U><B>Top</B></U></A><B> |</B></div> </td> </tr><tr> <td valign="top" rowspan="2" width=165 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 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>Ejercicios 5.1</B></div> <div> <B>La integral indefinida</B></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="0" CELLPADDING="4" CELLSPACING="1" WIDTH="903" BORDERCOLORLIGHT="#006666" BORDERCOLORDARK="#006666" FRAME="VOID" RULES="NONE" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#006699 HEIGHT= 23 WIDTH="885"><div> <B> <A NAME="apuntes"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Apuntes: Definiciones, t</B><B>e</B><B>oremas.</B></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 45 WIDTH="442"><div> <A HREF="#antiderivada" TITLE="5.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Definición 5.1.1</U></A>: Antiderivada</div> </tD> <tD VALIGN=TOP WIDTH="442"><div> <A HREF="#teorema_5_1_1" TITLE="5.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 5.1.1</U></A>:  Las antiderivadas difieren por una constante</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 45 ><NOBR><div> <A HREF="#integral_indefinida" TITLE="5.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Notación de la integral indefinida</U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="442"><div> <A HREF="#tabla" TITLE="5.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Tabla 5.1.1</U></A>: Derivadas y fórmulas de integración correspondientes</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 45 WIDTH="442"><div> <A HREF="#teorema_5_1_2" TITLE="5.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 5.1.2</U></A>:  Propiedades de la integral indefinida</div> </tD> <tD VALIGN=TOP WIDTH="442"><div> <A HREF="#ecuación_diferencial" TITLE="5.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ecuación diferencial de primer orden que se resuelve por integración indefinida</U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#CCFFFF HEIGHT= 45 WIDTH="442"><div> <A HREF="#ejercicios_5_1" TITLE="5.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ejercicios 5.1</U></A>: Enunciados y soluciones de los problemas</div> </tD> <tD VALIGN=TOP BGCOLOR=#CCFFCC WIDTH="442"><div> <A HREF="#lista_de_reproducción" TITLE="5.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ejercicios 5.1</U></A>: Lista de reproduccción de los videos en You Tube</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#006699 HEIGHT= 20 ><NOBR><div> <U>Nota</U>: Las explicaciones magistrales de esta teoría las puede leer en el libro "Cálculo", Trascendentes tempranas, Edi. 4, del Profesor D.G. Zill </div> </NOBR></tD> </tR> </TABLE></div> <div>  <A NAME="antiderivada"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Antiderivada</div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">De</B><B>fin</B><B>ición 5.1.1:  </B>Antiderivada</div> <div> Se dice que una función <I>F</I> es una <I><B>antiderivada</B></I> de una función <I>f </I>sobre algún intervalo <I>I</I> si </div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="2c196c190.gif" border=0 width="108" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> para toda <I>x</I> en <I>I</I>.</div> <div>  <A NAME="teorema_5_1_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 5.1.1</div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 5.1.1:  </B>Las antiderivadas difieren por una constante</div> <div> Si <I>G </I>'(<I>x</I>) = <I>F </I>'(<I>x</I>) para toda <I>x</I> en algún intervalo [<I>a</I>, <I>b</I>], entonces</div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="2c1b6c190.gif" border=0 width="108" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> para toda <I>x</I> en el intervalo.</div> <div>  <A NAME="integral_indefinida"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>integral indefinida</div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Notación de la integral indefinida</B></div> <div> Si <I>F </I>'(<I>x</I>) = <I>f </I>(<I>x</I>), la antiderivada más general de <I>f</I> se representa por: </div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="2c1d93240.gif" border=0 width="147" height="36" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="2c1e74df0.gif" border=0 width="884" height="223" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="tabla"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Tabla</div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Tabla 5.1.1: </B>Derivadas y fórmulas de integración correspondientes</div> <div> Siempre que se obtiene la derivada de una función, al mismo tiempo se obtiene una fórmula de integración correspondiente. De esta manera se construye la siguiente tabla de integrales básicas:</div> <bR> <div> <TABLE BORDER="2" CELLPADDING="2" CELLSPACING="1" WIDTH="911" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 59 WIDTH="446"><div> <IMG SRC="2c20df340.gif" border=0 width="223" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="2c28bf340.gif" border=0 width="447" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 62 ><NOBR><div> <IMG SRC="2c2175370.gif" border=0 width="373" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="2c29b4370.gif" border=0 width="436" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 62 WIDTH="446"><div> <IMG SRC="2c2229340.gif" border=0 width="297" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="2c2a97340.gif" border=0 width="407" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 62 ><NOBR><div> <IMG SRC="2c2363340.gif" border=0 width="355" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="2c2bc0370.gif" border=0 width="448" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 59 ><NOBR><div> <IMG SRC="2c247c340.gif" border=0 width="380" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="2c2c50340.gif" border=0 width="336" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 59 ><NOBR><div> <IMG SRC="2c2564340.gif" border=0 width="356" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="446"><div> <IMG SRC="2c2d17340.gif" border=0 width="279" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 59 ><NOBR><div> <IMG SRC="2c267c340.gif" border=0 width="380" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="2c2e99340.gif" border=0 width="409" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 59 ><NOBR><div> <IMG SRC="2c27ac340.gif" border=0 width="428" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="2c2f99340.gif" border=0 width="409" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <B> <A NAME="teorema_5_1_2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Teorema 5.1.2</B></div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 5.1.2:</B>  Propiedades de la integral indefinida</div> <div> <IMG SRC="2c3d3f620.gif" border=0 width="575" height="98" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="ecuación_diferencial"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ecuación diferencial</div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ecuación diferencial de primer orden que se resuelve por integración indefinida</B></div> <div> En términos generales, una ecuación diferencial es una ecuación que implica las derivadas o el diferencial de una función desconocida. </div> <div> Las ecuaciones diferenciales se clasifican según el <I><B>orden</B></I> de la derivada más alta que aparece en la ecuación.</div> <div> Resolver una ecuación diferencial consiste en hallar una familia de funciones que satisfacen la ecuación.</div> <div> Una <I><B>ecuación diferencial de primer orden</B></I> de la forma:</div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="2c3150300.gif" border=0 width="80" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> puede resolverse usando integración indefinida:</div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="2c339d300.gif" border=0 width="669" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="ejercicios_5_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ejercicios 5.1</div> <div> <B>Enunciados y soluciones de los problemas de los Ejercicios 5.1</B></div> <bR> <div> <FONT SIZE="5" COLOR="#006600" FACE="Bell MT" ><B>S</B></FONT>olución en imagen <IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> y o video <FONT SIZE="3" COLOR="#330033" FACE="Times New Roman" ><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT> de los problemas de los Ejercicios 5.1:</div> <bR> <div> <FONT SIZE="4" COLOR="#006666" FACE="Bell MT" ><B>E</B></FONT>n los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >1 </FONT>a<FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" > 30</FONT>, evalúe la integral indefinida dada.</div> <bR> <div> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><IMG SRC="2bc145340.gif" border=0 width="69" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="https://youtu.be/rsfDuJhVNDQ" TARGET="_top" TITLE="https://youtu.be/rsfDuJhVNDQ"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >       <IMG SRC="2bc279340.gif" border=0 width="121" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="https://youtu.be/rsfDuJhVNDQ?t=1m11s" TARGET="_top" TITLE="https://youtu.be/rsfDuJhVNDQ?t=1m11s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >       <IMG SRC="2bc34d340.gif" border=0 width="77" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="https://youtu.be/rsfDuJhVNDQ?t=1m44s" TARGET="_top" TITLE="https://youtu.be/rsfDuJhVNDQ?t=1m44s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >       </FONT><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><IMG SRC="2bc460340.gif" border=0 width="96" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="https://youtu.be/rsfDuJhVNDQ?t=2m54s" TARGET="_top" TITLE="https://youtu.be/rsfDuJhVNDQ?t=2m54s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >        <IMG SRC="2bc55b340.gif" border=0 width="91" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="https://youtu.be/rsfDuJhVNDQ?t=5m9s" TARGET="_top" TITLE="https://youtu.be/rsfDuJhVNDQ?t=5m9s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >        <IMG SRC="2bc65d340.gif" border=0 width="93" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="https://youtu.be/rsfDuJhVNDQ?t=7m44s" TARGET="_top" TITLE="https://youtu.be/rsfDuJhVNDQ?t=7m44s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" > </FONT></div> <div> <IMG SRC="2bc787340.gif" border=0 width="135" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#003333" FACE="Bell MT" ><A HREF="https://youtu.be/hyRxSxpMdC0?t=20s" TARGET="_top" TITLE="https://youtu.be/hyRxSxpMdC0?t=20s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="2bc879340.gif" border=0 width="121" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#003333" FACE="Bell MT" ><A HREF="https://youtu.be/hyRxSxpMdC0?t=2m46s" TARGET="_top" TITLE="https://youtu.be/hyRxSxpMdC0?t=2m46s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="2bc9a4340.gif" border=0 width="164" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#003333" FACE="Bell MT" ><A HREF="https://youtu.be/hyRxSxpMdC0?t=5m16s" TARGET="_top" TITLE="https://youtu.be/hyRxSxpMdC0?t=5m16s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="2bcab1340.gif" border=0 width="177" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#003333" FACE="Bell MT" ><A HREF="https://youtu.be/hyRxSxpMdC0?t=7m20s" TARGET="_top" TITLE="https://youtu.be/hyRxSxpMdC0?t=7m20s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="2bcb9f340.gif" border=0 width="159" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/XE75rdEWgbE?t=21s" TARGET="_top" TITLE="https://youtu.be/XE75rdEWgbE?t=21s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2bccaf370.gif" border=0 width="175" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/XE75rdEWgbE?t=3m48s" TARGET="_top" TITLE="https://youtu.be/XE75rdEWgbE?t=3m48s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2bcd8b340.gif" border=0 width="139" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/XE75rdEWgbE?t=7m34s" TARGET="_top" TITLE="https://youtu.be/XE75rdEWgbE?t=7m34s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2bce8f340.gif" border=0 width="143" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/XE75rdEWgbE?t=8m55s" TARGET="_top" TITLE="https://youtu.be/XE75rdEWgbE?t=8m55s"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c3490340.gif" border=0 width="144" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="#15" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <FONT SIZE="4" COLOR="#003333" FACE="Bell MT" ><IMG SRC="2c38c8340.gif" border=0 width="200" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#16" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c3aa4340.gif" border=0 width="164" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#17" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c4085390.gif" border=0 width="133" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#18" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c44b8340.gif" border=0 width="184" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#19" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2c4790390.gif" border=0 width="144" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#20" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c4ad5340.gif" border=0 width="213" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#21" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c4be3340.gif" border=0 width="227" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#22" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c4edf340.gif" border=0 width="223" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#23" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2c5079340.gif" border=0 width="121" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#24" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c52a4340.gif" border=0 width="164" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#25" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c55af340.gif" border=0 width="175" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#26" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c57af340.gif" border=0 width="175" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#27" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2c59d3340.gif" border=0 width="211" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#28" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c5ccb340.gif" border=0 width="203" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#29" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c6578340.gif" border=0 width="120" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#30" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>      </div> <bR> <div> <FONT SIZE="4" COLOR="#006666" FACE="Bell MT" ><B>E</B></FONT>n los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >31 </FONT>y<FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" > 32</FONT>, use una identidad trigonométrica para hallar la integral indefinida indicada.</div> <div> <IMG SRC="2c6878340.gif" border=0 width="120" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#31" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2c6b7f340.gif" border=0 width="127" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#32" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> <FONT SIZE="4" COLOR="#006666" FACE="Bell MT" ><B>E</B></FONT>n los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" >33 </FONT>a<FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" > 40</FONT>, use diferenciación y la regla de la cadena para comprobar el resultado de integración dado.</div> <div> <IMG SRC="2c6def340.gif" border=0 width="239" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#33" TITLE="5.1"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <DIV ALIGN="LEFT"></DIV> <div>  <A NAME="lista_de_reproducción"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Lista de reproducción</div> <div> <B>Lista de reproducción de los Ejercicios 5.1</B></div> <bR> <div> <B> <iframe width="853" height="480" src="https://www.youtube.com/embed/videoseries?list=PLq_SQwoOZHo0k-Y2ZDZ-IPpBKNyAddgOA" frameborder="0" allowfullscreen></iframe>  </B></div> <div> <B> <A NAME="15"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>15</B></div> <div> <IMG SRC="2c3650110.gif" border=0 width="848" height="273" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="16"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>16</div> <div> <IMG SRC="2c398c140.gif" border=0 width="908" height="276" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="17"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>17</div> <div> <IMG SRC="2c3ee0350.gif" border=0 width="992" height="309" ALT="MathType 6.0 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>18</div> <div> <IMG SRC="2c42958b0.gif" border=0 width="917" height="395" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="19"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>19</div> <div> <IMG SRC="2c4527750.gif" border=0 width="807" height="373" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="20"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>20</div> <div> <IMG SRC="2c48f14f0.gif" border=0 width="1009" height="335" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="21"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>21</div> <div> <IMG SRC="2c495c700.gif" border=0 width="860" height="368" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="22"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>22</div> <div> <IMG SRC="2c4c4c370.gif" border=0 width="844" height="311" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="23"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>23</div> <div> <IMG SRC="2c4db8340.gif" border=0 width="696" height="308" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="24"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>24</div> <div> <IMG SRC="2c4f638c0.gif" border=0 width="867" height="140" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="25"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>25</div> <div> <IMG SRC="2c51cfc10.gif" border=0 width="719" height="193" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="26"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>26</div> <div> <IMG SRC="2c545f890.gif" border=0 width="863" height="137" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="27"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>27</div> <div> <IMG SRC="2c56a4890.gif" border=0 width="676" height="137" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="28"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>28</div> <div> <IMG SRC="2c5875c10.gif" border=0 width="629" height="193" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="29"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>29</div> <div> <IMG SRC="2c5b80c10.gif" border=0 width="640" height="193" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0">      <IMG SRC="2c634ba00.gif" border=0 width="331" height="160" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="30"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>30</div> <div> <IMG SRC="2c6488c10.gif" border=0 width="648" height="193" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0">     <IMG SRC="2c671eb40.gif" border=0 width="286" height="180" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="31"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>31</div> <div> <IMG SRC="2c695c110.gif" border=0 width="860" height="273" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="32"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>32</div> <div> <IMG SRC="2c6a6cac0.gif" border=0 width="876" height="428" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="33"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>33</div> <div> <IMG SRC="2c6e87490.gif" border=0 width="903" height="329" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <bR> <DIV ALIGN="LEFT"></DIV> <div> <B> <img src="HTMLobj-11478/aniGif.gif" border="0" align="bottom" width="550" height="14">  </B></div> <bR> <div> <IMG SRC="2c128c8c0.gif" border=0 width="140" height="140" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <A HREF="http://twitter.com/Calculo21" TARGET="_top" TITLE="http://twitter.com/Calculo21"><U><IMG SRC="248328280.jpg" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="http://www.facebook.com/juan.beltran.58511" TARGET="_top" TITLE="http://www.facebook.com/juan.beltran.585"><U><IMG SRC="248428280.gif" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="https://www.youtube.com/user/MatematicaAplicadas" TARGET="_top" TITLE="https://www.youtube.com/user/MatematicaA"><U><IMG SRC="248528280.jpg" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div>  <a href="http://twitter.com/Calculo21" class="twitter-follow-button" data-button="grey" data-text-color="#FFFFFF" data-link-color="#00AEFF" data-show-count="false" data-lang="es">Follow @Calculo21</a> <script src="http://platform.twitter.com/widgets.js" type="text/javascript"></script>  </div> <bR> <div> Por: <FONT SIZE="4" COLOR="#FF9900" FACE="Times New Roman" >Juan Carlos </FONT><FONT SIZE="4" COLOR="#0000FF" FACE="Times New Roman" >Beltrán</FONT> <FONT SIZE="4" COLOR="#FF0000" FACE="Times New Roman" >Beltrán </FONT></div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:970px;height:250px" data-ad-client="ca-pub-6266400581769473" data-ad-slot="7846235711"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <A HREF="http://jcbmat.com/" TARGET="_top" TITLE="http://jcbmat.com/"><U><B><IMG SRC="2cd2e5530.gif" border=0 width="741" height="83" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> | <A HREF="http://www.aritmetica.jcbmat.com/" TARGET="_top" TITLE="http://www.aritmetica.jcbmat.com/"><U>Aritmética</U></A> | <A HREF="http://www.algebra.jcbmat.com/index.htm" TARGET="_top" TITLE="http://www.algebra.jcbmat.com/index.htm"><U>Álgebra de Baldor</U></A> | <A HREF="http://www.precalculo.jcbmat.com/" TARGET="_top" TITLE="http://www.precalculo.jcbmat.com/"><U>Precálculo</U></A> |</div> <div> | <A HREF="http://www.calculo.jcbmat.com/" TARGET="_top" TITLE="http://www.calculo.jcbmat.com/"><U>Cálculo de una variable</U></A> | </div> <div> | <A HREF="http://www.ecuacionesdiferenciales.jcbmat.com/index.htm" TARGET="_top" TITLE="http://www.ecuacionesdiferenciales.jcbma"><U>Ecuaciones diferenciales</U></A> | <A HREF="http://www.matu.jcbmat.com/id1383.htm" TARGET="_top" TITLE="http://www.matu.jcbmat.com/id1383.htm"><U>Análisis complejo</U></A> |</div> <div> | <A HREF="http://www.algebralineal.jcbmat.com/index.htm" TARGET="_top" TITLE="http://www.algebralineal.jcbmat.com/inde"><U>Álgebra lineal</U></A> | <A HREF="http://www.jcbmat.com/id63.htm" TARGET="_top" TITLE="http://www.jcbmat.com/id63.htm"><U>Administración y economía</U></A> | <A HREF="http://www.geometrianalitica.jcbmat.com/" TARGET="_top" TITLE="http://www.geometrianalitica.jcbmat.com/"><U>Geometría analítica</U></A> |  </div> <bR> <div> | <A HREF="http://calculo21.blogspot.com/" TARGET="_top" TITLE="http://calculo21.blogspot.com/"><U>Blog Cálculo21</U></A> | <A HREF="http://videomate21.blogspot.com/" TARGET="_top" TITLE="http://videomate21.blogspot.com/"><U>Blog Algebra de Baldor</U></A> | <A HREF="http://equat21z.blogspot.com/" TARGET="_top" TITLE="http://equat21z.blogspot.com/"><U>Blog Ecuaciones diferenciales</U></A> | <A HREF="http://bancomate.blogspot.com.co/" TARGET="_top" TITLE="http://bancomate.blogspot.com.co/"><U>Blog Banco de matemáticas</U></A> |</div> <div> | <A HREF="https://www.youtube.com/user/MatematicaAplicadas" TARGET="_top" TITLE="https://www.youtube.com/user/MatematicaA"><U>Canal de YouTube</U></A> |</div> <bR> </td> <td valign="top" rowspan="2" width=105 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <A HREF="https://www.youtube.com/user/MatematicaAplicadas" TARGET="_top" TITLE="https://www.youtube.com/user/MatematicaA"><U><B><IMG SRC="2c0564580.gif" border=0 width="100" height="600" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <bR> <bR> </td> </tr><tr> <td valign="top" height=24 BGCOLOR="#006666" TEXT="#080000"> <div>  <form action="http://www.google.com.co" id="cse-search-box" target="_blank"> <div> <input type="hidden" name="cx" value="partner-pub-6266400581769473:6497222116" /> <input type="hidden" name="ie" value="UTF-8" /> <input type="text" name="q" size="64" /> <input type="submit" name="sa" value="Buscar en MatU" /> </div> </form> <script type="text/javascript" src="http://www.google.com.co/coop/cse/brand?form=cse-search-box&lang=es"></script>  </div> </td> </tr></table></body>