5.2

<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="id548_m.htm#apuntes" TARGET="TRLX_Middle" TITLE="5.2"><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 5.2</B></div> <div> <B>Integración por sustitución </B><I><B>u</B></I></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= 23 ><NOBR><div> <A HREF="#teorema_5_2_1" TITLE="5.2"><U><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 5.2.1</U></A>: <B> </B>Regla de la sustitución <I>u</I></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#tabla_5_2_1" TITLE="5.2"><U><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Tabla 5.2.1</U></A>: Fórmulas de integración</div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#CCFFFF HEIGHT= 45 WIDTH="442"><div> <A HREF="#ejercicios_5_2" TITLE="5.2"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ejercicios 5.2</U></A>: Enunciados y soluciones de los problemas</div> </tD> <tD VALIGN=TOP BGCOLOR=#CCFFCC WIDTH="442"><div> <A HREF="#lista_de_reproducción_5_2" TITLE="5.2"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Ejercicios 5.2</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="teorema_5_2_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 5.2.1</div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 5.2.1:  </B>Regla de la sustitución <I>u</I></div> <div> Si u = g(x) es una función diferenciable cuyo rango es un intervalo I, f es una función continua sobre I y F es una antiderivada de f sobre I, entonces</div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="2c8bf1240.gif" border=0 width="241" height="36" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="tabla_5_2_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Tabla 5.2.1</div> <div> <B><IMG BORDER="0" SRC="Symb075c00b7011800660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Tabla 5.2.1: </B>Fórmulas de integración</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= 40 WIDTH="446"><div> <IMG SRC="2c9173210.gif" border=0 width="115" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="446"><div> <IMG SRC="2c99eb210.gif" border=0 width="235" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 58 WIDTH="446"><div> <IMG SRC="2c92e0310.gif" border=0 width="224" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="446"><div> <IMG SRC="2c9a28330.gif" border=0 width="296" height="51" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 62 WIDTH="446"><div> <IMG SRC="2c939f300.gif" border=0 width="159" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="446"><div> <IMG SRC="2c9b24310.gif" border=0 width="292" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 59 WIDTH="446"><div> <IMG SRC="2c94b8210.gif" border=0 width="184" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="2c9c41340.gif" border=0 width="321" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 56 WIDTH="446"><div> <IMG SRC="2c95c5210.gif" border=0 width="197" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="446"><div> <IMG SRC="2c9da4310.gif" border=0 width="164" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 59 WIDTH="446"><div> <IMG SRC="2c96b4210.gif" border=0 width="180" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="446"><div> <IMG SRC="2c9e17340.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= 40 WIDTH="446"><div> <IMG SRC="2c97c1210.gif" border=0 width="193" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="446"><div> <IMG SRC="2c9fd8210.gif" border=0 width="216" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 40 WIDTH="446"><div> <IMG SRC="2c98dd210.gif" border=0 width="221" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="446"><div> <IMG SRC="2ca0c9210.gif" border=0 width="201" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 40 WIDTH="446"><div> <IMG SRC="2cedc9210.gif" border=0 width="201" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="446"><div> <IMG SRC="2d0515210.gif" border=0 width="277" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 40 WIDTH="446"><div> <IMG SRC="2cddc9210.gif" border=0 width="201" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="446"><div> <IMG SRC="2d040d210.gif" border=0 width="269" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> </TABLE></div> <DIV ALIGN="LEFT"></DIV> <div>  <A NAME="ejercicios_5_2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Ejercicios 5.2</div> <div> <B>Enunciados y soluciones de los problemas de los Ejercicios 5.2</B></div> <bR> <div> <B>S</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >olución en imagen <IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> y o video <IMG SRC="2c85181a0.jpg" border=0 width="24" height="26" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> de los problemas de los Ejercicios 5.2:</FONT></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" > 50</FONT>, evalúe la integral indefinida dada usando una sustitución <I>u</I> idónea.</div> <bR> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><IMG SRC="2cfc73210.gif" border=0 width="115" height="33" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#1" TITLE="5.2"><U><IMG SRC="265318180.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="2d07882c0.gif" border=0 width="136" height="44" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#2" TITLE="5.2"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2d0880380.gif" border=0 width="128" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="https://youtu.be/N_aqbwJdsoY" TARGET="_top" TITLE="https://youtu.be/N_aqbwJdsoY"><U><IMG SRC="2c85181a0.jpg" border=0 width="24" height="26" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2d097d310.gif" border=0 width="125" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="https://youtu.be/N_aqbwJdsoY?t=3m50s" TARGET="_top" TITLE="https://youtu.be/N_aqbwJdsoY?t=3m50s"><U><IMG SRC="2c85181a0.jpg" border=0 width="24" height="26" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2d0a7f310.gif" border=0 width="127" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="https://youtu.be/N_aqbwJdsoY?t=5m52s" TARGET="_top" TITLE="https://youtu.be/N_aqbwJdsoY?t=5m52s"><U><IMG SRC="2c85181a0.jpg" border=0 width="24" height="26" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2d0b74330.gif" border=0 width="116" height="51" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="https://youtu.be/N_aqbwJdsoY?t=9m2s" TARGET="_top" TITLE="https://youtu.be/N_aqbwJdsoY?t=9m2s"><U><IMG SRC="2c85181a0.jpg" border=0 width="24" height="26" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <DIV ALIGN="LEFT"></DIV> <div>  <A NAME="lista_de_reproducción_5_2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Lista de reproducción 5.2</div> <div> <B>Lista de reproducción de los Ejercicios 5.2</B></div> <bR> <div> <B> <iframe width="853" height="480" src="https://www.youtube.com/embed/videoseries?list=PLq_SQwoOZHo0FEhHsg7x5eaFSv0vwxiY-" frameborder="0" allowfullscreen></iframe>  </B></div> <div>  <A NAME="1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1</div> <div> <IMG SRC="2d03592f0.gif" border=0 width="601" height="303" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>2</div> <div> <IMG SRC="2d068c630.gif" border=0 width="652" height="355" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <bR> <bR> <DIV ALIGN="LEFT"></DIV> <div> <B> <img src="HTMLobj-11398/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 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