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=24 BGCOLOR="#006666" TEXT="#080000"> <div> <B>C á l c u l o   2 1</B><FONT SIZE="4" COLOR="#FFCC66" FACE="Bell MT" ><B> </B></FONT><FONT SIZE="4" COLOR="#FFFFCC" FACE="Bell MT" >   </FONT><B> </B> |   <A HREF="index.htm" TARGET="_top" TITLE="Cálculo 21"><U>Home</U></A>  |  <A HREF="id98.htm" TARGET="_top" TITLE="Contenido"><U>Contenido</U></A>  |  <A HREF="id341.htm" TARGET="_top" TITLE="Introducción al cálculo integral"><U>Cálculo integral</U></A>  |  <A HREF="id466.htm" TARGET="_top" TITLE="Técnicas de integración"><U>Técnicas de integración</U></A>  |</div> </td> </tr><tr> <td valign="top" rowspan="2" width=160 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B> <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3073775660"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </B></div> </td> <td valign="top" height=432 colspan=2 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> Integración por partes</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>  <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="21bb84130.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="21bc451d0.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="21bd491d0.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="21be581d0.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="21bf4f1d0.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="21c0791d0.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="21c1591d0.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="21c2581d0.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="21c3601d0.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="21c45b1d0.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="21c5711d0.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="21c67d1d0.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="21c7641d0.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="21c8e9d50.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="21c9e9dd0.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="21ca60d10.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="21cbedc00.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="21cc2fef0.gif" border=0 width="559" height="239" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><IMG SRC="21cd442b0.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="21ce43150.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="21cfe9e90.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="21d0e99d0.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="21d15b1d0.gif" border=0 width="91" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21d2d41d0.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="21d3711d0.gif" border=0 width="113" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21d4e9b80.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="21d57d1d0.gif" border=0 width="125" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21d6fb200.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="21d7641d0.gif" border=0 width="100" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="21d8e0b30.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=24 BGCOLOR="#006666" TEXT="#080000"> <div> <B> <form action="http://www.google.com.co" id="cse-search-box" target="_blank"> <div> <input type="hidden" name="cx" value="partner-pub-6266400581769473:4577786115" /> <input type="hidden" name="ie" value="UTF-8" /> <input type="text" name="q" size="100" /> <input type="submit" name="sa" value="Buscar" /> </div> 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