Integración por partes

<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> <A HREF="id354.htm" TARGET="_top" TITLE="Integración directa"><U>Integración directa</U></A>   |   <A HREF="id355.htm" TARGET="_top" TITLE="Integración por sustitución"><U>Integración por sustitución</U></A>   |   Integración por partes   |   <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=112 BGCOLOR="#006600" TEXT="#F7F7FF"> <bR> <bR> <div> <A HREF="id356_m.htm#arriba" TARGET="TRLX_Middle" TITLE="Integración por partes"><U><B> <img src="HTMLobj-5980/aniGif.gif" alt="Arriba" border="0" align="bottom" width="50" height="50">  </B></U></A></div> </td> <td valign="top" height=361 colspan=2 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <U>Integración por partes</U></div> <div>  <A NAME="arriba"><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>L</B></FONT>a fórmula para la "integración por partes", se deduce a partir de la regla de la derivada de un producto de funciones. Veamos:</div> <div> <IMG SRC="173e84130.gif" border=0 width="644" height="275" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div>  <object width="425" height="344"><param name="movie" value="http://www.youtube.com/v/LYSqub0s3c4&hl=es&fs=1"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/LYSqub0s3c4&hl=es&fs=1" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="344"></embed></object>  </div> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="537" BORDERCOLORLIGHT="#33FFCC" BORDERCOLORDARK="#009900" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=3 BGCOLOR=#FFFFCC HEIGHT= 39 ><NOBR><div>  <A NAME="ejercicios_resueltos_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>Ejercicios resueltos</B></FONT></div> <div> <B>          E</B><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >n los ejercicios siguientes efectúe la integral indefinida:</FONT></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 WIDTH="173"><div> <A HREF="#1" TITLE="Integración por partes"><U><IMG SRC="173f451d0.gif" border=0 width="69" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="173"><div> <A HREF="#2_1" TITLE="Integración por partes"><U><IMG SRC="1740491d0.gif" border=0 width="73" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="173"><div> <A HREF="#3" TITLE="Integración por partes"><U><IMG SRC="1741581d0.gif" border=0 width="88" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 WIDTH="173"><div> <A HREF="#4" TITLE="Integración por partes"><U><IMG SRC="17424f1d0.gif" border=0 width="79" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#5" TITLE="Integración por partes"><U><IMG SRC="1743791d0.gif" border=0 width="121" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="173"><div> <A HREF="#6" TITLE="Integración por partes"><U><IMG SRC="1744591d0.gif" border=0 width="89" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 WIDTH="173"><div> <A HREF="#7" TITLE="Integración por partes"><U><IMG SRC="1745581d0.gif" border=0 width="88" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="173"><div> <A HREF="#8" TITLE="Integración por partes"><U><IMG SRC="1746601d0.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 WIDTH="173"><div> <A HREF="#9_1" TITLE="Integración por partes"><U><IMG SRC="17475b1d0.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= 36 WIDTH="173"><div> <A HREF="#10_1_1" TITLE="Integración por partes"><U><IMG SRC="1748711d0.gif" border=0 width="113" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#11_1" TITLE="Integración por partes"><U><IMG SRC="17497d1d0.gif" border=0 width="125" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="173"><div> <A HREF="#12_1" TITLE="Integración por partes"><U><IMG SRC="174a641d0.gif" border=0 width="100" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=3 HEIGHT= 66 WIDTH="525"><div>  <object width="425" height="344"><param name="movie" value="http://www.youtube.com/v/hSzENmw1hlk&hl=en&fs=1"></param><param name="allowFullScreen" value="true"></param><embed src="http://www.youtube.com/v/hSzENmw1hlk&hl=en&fs=1" type="application/x-shockwave-flash" allowfullscreen="true" width="425" height="344"></embed></object>  </div> </tD> </tR> </TABLE></div> <div> <B>S o l u c i o n e s</B></div> <div> <B> <A NAME="1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>1.</B><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >Solución:</FONT></div> <div> <IMG SRC="174be9d50.gif" border=0 width="489" height="213" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="2_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>2.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT>Solución:</div> <div> <IMG SRC="174ce9dd0.gif" border=0 width="489" height="221" 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="174d60d10.gif" border=0 width="608" height="465" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" > <A NAME="4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></FONT><B> </B><FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>4.</B></FONT><B> </B><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >Solución:</FONT></div> <div> <IMG SRC="174eedc00.gif" border=0 width="493" height="192" 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="174f2fef0.gif" border=0 width="559" height="239" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><IMG SRC="1750442b0.gif" border=0 width="68" height="43" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <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="175143150.gif" border=0 width="579" height="277" 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>Solución:</div> <div> <IMG SRC="1752e9e90.gif" border=0 width="489" height="233" 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="1753e99d0.gif" border=0 width="489" height="413" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="9_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="17545b1d0.gif" border=0 width="91" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="1755d41d0.gif" border=0 width="468" height="285" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="10_1_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="1756711d0.gif" border=0 width="113" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="1757e9b80.gif" border=0 width="489" height="184" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="11_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="17587d1d0.gif" border=0 width="125" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="1759fb200.gif" border=0 width="507" height="288" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="12_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="175a641d0.gif" border=0 width="100" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="175be0b30.gif" border=0 width="480" height="179" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> </td> </tr><tr> <td valign="top" colspan=2 height=21 BGCOLOR="#006600" TEXT="#F7F7FF"> <div> |  <A HREF="id356_m.htm#ejercicios_resueltos_1" TARGET="TRLX_Middle" TITLE="Integración por partes"><U><B>Ejercicios resueltos</B></U></A> |</div> </td> </tr></table></body>