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=98 BGCOLOR="#006600" TEXT="#F7F7FF" bgproperties="fixed" background="05700732dwnpjk.jpg"> <div> <FONT SIZE="6" COLOR="#FFFFFF" FACE="Times New Roman" ><B>C á l c u l o   2 1</B></FONT><FONT SIZE="6" COLOR="#FFCC66" FACE="Arial" ><B> </B></FONT><FONT SIZE="2" COLOR="#FFFFCC" FACE="Arial" >   </FONT><B>  |   </B><A HREF="index.htm" TARGET="_top" TITLE="A n á l i s i s M a t e m á t i c o"><U><B>Home</B></U></A><B> |  </B><A HREF="id98.htm" TARGET="_top" TITLE=" | Contenido |"><U><B>Contenido</B></U></A><B> |  </B><A HREF="id341.htm" TARGET="_top" TITLE="Introducción al cálculo integral"><U><B>Cálculo integral</B></U></A><B> |</B></div> <div> Integración directa   |   <A HREF="id355.htm" TARGET="_top" TITLE="Integración por sustitución"><U>Integración por sustitución</U></A>   |   <A HREF="id356.htm" TARGET="_top" TITLE="Integración por partes"><U>Integración por partes</U></A>   |   <A HREF="id357.htm" TARGET="_top" TITLE="Potencias de las funciones trigonométric"><U>Potencias de las funciones trigonométricas</U></A>   |   <A HREF="id358.htm" TARGET="_top" TITLE="Sustitución trigonométrica"><U>Sustitución trigonométrica</U></A>   |   <A HREF="id359.htm" TARGET="_top" TITLE="Integración de funciones racionales, por"><U>Integración de funciones racionales, por fracciones parciales, cuando el denominador sólo tiene factores lineales</U></A>   |   <A HREF="id360.htm" TARGET="_top" TITLE="Integración de funciones racionales, por"><U>Integración de funciones racionales, por fracciones parciales, cuando el denominador contiene factores cuadráticos</U></A>   |   <A HREF="id361.htm" TARGET="_top" TITLE="Integrales en las que aparecen expresion"><U>Integrales en las que aparecen expresiones cuadráticas</U></A>   |   <A HREF="id362.htm" TARGET="_top" TITLE="Integrales que producen funciones trigon"><U>Integrales que producen funciones trigonométricas inversas</U></A>   |   <A HREF="id363.htm" TARGET="_top" TITLE="Misceláneas"><U>Misceláneas</U></A></div> <div> </div> </td> </tr><tr> <td valign="top" rowspan="2" width=245 BGCOLOR="#006600" TEXT="#F7F7FF"> <div> <A HREF="id354_m.htm#propiedades_fundamentales_de_la" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B>Propiedades</B></U></A></div> <div> <A HREF="id354_m.htm#p1_1" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad1</B></U></A></div> <div> <A HREF="id354_m.htm#p2" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad2</B></U></A></div> <div> <A HREF="id354_m.htm#p3" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad3</B></U></A></div> <div> <A HREF="id354_m.htm#p4" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad4</B></U></A></div> <div> <A HREF="id354_m.htm#p5a" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad5</B></U></A></div> <div> <A HREF="id354_m.htm#p6" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad6</B></U></A></div> <div> <A HREF="id354_m.htm#p7" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad7</B></U></A></div> <div> <A HREF="id354_m.htm#p8" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad8</B></U></A></div> <div> <A HREF="id354_m.htm#p9" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad9</B></U></A></div> <div> <A HREF="id354_m.htm#p10" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad10</B></U></A></div> <div> <A HREF="id354_m.htm#p11" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad11</B></U></A></div> <div> <A HREF="id354_m.htm#p12" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad12</B></U></A></div> <div> <A HREF="id354_m.htm#p13" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad13</B></U></A></div> <div> <A HREF="id354_m.htm#p14" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad14</B></U></A></div> <div> <A HREF="id354_m.htm#p15" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad15</B></U></A></div> <div> <A HREF="id354_m.htm#p16" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad16</B></U></A></div> <div> <A HREF="id354_m.htm#p17" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad17</B></U></A></div> <div> <A HREF="id354_m.htm#p18" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad18</B></U></A></div> <div> <A HREF="id354_m.htm#p19" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad19</B></U></A></div> <div> <A HREF="id354_m.htm#p20" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad20</B></U></A></div> <div> <A HREF="id354_m.htm#p21" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad21</B></U></A></div> <div> <A HREF="id354_m.htm#p22" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad22</B></U></A></div> <div> <A HREF="id354_m.htm#p23" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad23</B></U></A></div> <div> <A HREF="id354_m.htm#p24" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad24</B></U></A></div> <div> <A HREF="id354_m.htm#25" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B><IMG BORDER="0" SRC="Wing0a73007700a0ffffff00.gif" HEIGHT="11" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Propiedad25</B></U></A></div> <bR> </td> <td valign="top" height=361 colspan=2 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <U>Integración directa</U></div> <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="16bd0c1d0.gif" border=0 width="268" height="29" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <DIV ALIGN="LEFT"></DIV> <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="16be3c1d0.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="16bf90380.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="16c037500.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="16c1eb350.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="16c2691d0.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="16c3dd3d0.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= 46 ><NOBR><div>  <A NAME="p7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="16c4d1270.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="16c57f1d0.gif" border=0 width="127" height="29" ALT="MathType 5.0 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="16c6b41d0.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="16c7ef3d0.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="16c8b41d0.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="16c9a81d0.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="16cab01d0.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="16cbbb1d0.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="16ccc91d0.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="16cdd41d0.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="16cec82d0.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="16cfbd280.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="16d0d12d0.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="16d1b81d0.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="16d2b81d0.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="16d3c01d0.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="16d4c91d0.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="16d5eb1d0.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="16d6eb1d0.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="16d7601d0.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="16d8a71d0.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="16d9912b0.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="16da9b1d0.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="16db6d1d0.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="16dc7b280.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="16dd80270.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="16de4d2b0.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="16df60280.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="16e050230.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="16e18f6d0.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="16e2545d0.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="16e375c50.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="16e4b3a80.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="16e5bbe40.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="16e615d30.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="16e72fcc0.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="16e84d2b0.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="16e9ac700.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="16ea60280.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="16eb61690.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="16ec50230.gif" border=0 width="80" height="35" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B><IMG SRC="16ede7430.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=21 BGCOLOR="#006600" TEXT="#F7F7FF"> <div> <B>|  </B><A HREF="id354_m.htm#arriba" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B>Arriba</B></U></A><B> |  </B><A HREF="id354_m.htm#ejercicios_resueltos" TARGET="TRLX_Middle" TITLE="Integración directa"><U><B>Ejercicios resueltos</B></U></A><B> |</B></div> </td> </tr></table></body>