Potencias de las funciones trigonométricas

<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><FONT SIZE="2" COLOR="#FFFFFF" FACE="Arial" ><A HREF="id471_m.htm#identidades_trigonom_tricas_2" TARGET="TRLX_Middle" TITLE="Potencias de las funciones trigonométric"><U><B>es</B></U></A></FONT></div> </td> <td valign="top" height=432 colspan=2 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> Potencias de las funciones trigonométricas</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>E</B></FONT>n este apartado aprenderemos a integrar funciones que presentan potencias trigonométricas, es decir, funciones con alguna de las siguientes formas:</div> <div> <IMG SRC="21dc7d180.gif" border=0 width="637" height="24" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> Para tal efecto es conveniente tener frescas en la memoria las siguientes identidades trigonométricas:</div> <DIV ALIGN="LEFT"></DIV> <div> <TABLE BORDER="2" CELLPADDING="2" CELLSPACING="1" WIDTH="617" BORDERCOLORLIGHT="#FFCC33" BORDERCOLORDARK="#996600" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=4 BGCOLOR=#FFCC66 HEIGHT= 20 WIDTH="593"><div>  <A NAME="identidades_trigonom_tricas_2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="3" COLOR="#006600" FACE="Times New Roman" ><B>Identidades trigonométricas</B></FONT></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 46 ><NOBR><div> <IMG SRC="21dd75150.gif" border=0 width="117" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="21de75150.gif" border=0 width="117" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="21df79270.gif" border=0 width="121" height="39" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="21e07c270.gif" border=0 width="124" height="39" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 28 ><NOBR><div> <IMG SRC="21e177150.gif" border=0 width="119" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="21e277150.gif" border=0 width="119" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP COLSPAN=2 WIDTH="295"><div> <IMG SRC="21e385130.gif" border=0 width="133" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 46 ><NOBR><div> <IMG SRC="21e423270.gif" border=0 width="291" height="39" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP COLSPAN=2><NOBR><div> <IMG SRC="21e52b270.gif" border=0 width="299" height="39" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div>      Por lo regular, una vez concluimos con las transformaciones trigonométricas adecuadas, el integrando queda expedito para aplicar la <A HREF="id469.htm" TARGET="_top" TITLE="Integración por sustitución"><U><I>integración por sustitución</I></U></A>. En otros casos debemos recurrir a la <A HREF="id470.htm" TARGET="_top" TITLE="Integración por partes"><U><I>integración por partes</I></U></A>.</div> <DIV ALIGN="LEFT"></DIV> <div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="1" WIDTH="633" BORDERCOLORLIGHT="#33FFCC" BORDERCOLORDARK="#009900" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=3 BGCOLOR=#00CCCC HEIGHT= 57 WIDTH="621"><div>  <A NAME="ejercicios_resueltos_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>Ejercicios resueltos</B></FONT></div> <bR> <div> <B>          </B><FONT SIZE="3" COLOR="#CCFF00" FACE="Times New Roman" ><B>E</B></FONT><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >n los siguientes ejercicios evalúe la integral indefinida:</FONT></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 36 WIDTH="205"><div> <A HREF="#1___2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21e6531d0.gif" border=0 width="83" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#2___2_1" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21e7831d0.gif" border=0 width="131" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#3___2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21e87c1d0.gif" border=0 width="124" 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="205"><div> <A HREF="#4___2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21e9531d0.gif" border=0 width="83" 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___2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21ea8b1d0.gif" border=0 width="139" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#6___2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21eb7d1d0.gif" border=0 width="125" 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="205"><div> <A HREF="#7___2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21ec7d1d0.gif" border=0 width="125" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#8___2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21ed541d0.gif" border=0 width="84" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#9___2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21ee541d0.gif" border=0 width="84" 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="205"><div> <A HREF="#10__2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21ef5c1d0.gif" border=0 width="92" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#11__2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21f0831d0.gif" border=0 width="131" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#12___2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21f1831d0.gif" border=0 width="131" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 42 WIDTH="205"><div> <A HREF="#13___2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21f25b1d0.gif" border=0 width="91" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#14___2" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21f3831d0.gif" border=0 width="131" height="29" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="205"><div> <A HREF="#15" TITLE="Potencias de las funciones trigonométric"><U><IMG SRC="21f485230.gif" border=0 width="133" height="35" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> </TABLE></div> <bR> <div> <B>S o l u c i o n e s </B></div> <bR> <div>  <A NAME="1___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>1. </B></FONT>Solución:</div> <div> <IMG SRC="21f523d70.gif" border=0 width="547" height="215" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="2___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="21f6afbd0.gif" border=0 width="431" height="189" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="3___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>3.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT>Solución:</div> <div> <IMG SRC="21f7533b0.gif" border=0 width="595" height="315" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="4___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>4.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT>Solución:</div> <div> <IMG SRC="21f81bab0.gif" border=0 width="539" height="171" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="5___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>5. </B></FONT>Solución:</div> <div> <IMG SRC="21f9497f0.gif" border=0 width="585" height="127" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="6___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>6.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT>Solución:</div> <div> <IMG SRC="21fa60000.gif" border=0 width="608" height="256" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" > <A NAME="7___2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></FONT><B>7</B>.<FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" > </FONT><FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" >Solución:</FONT></div> <div> <IMG SRC="21fb67000.gif" border=0 width="615" height="256" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="8___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>8.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT>Solución:</div> <div> <IMG SRC="21fc48170.gif" border=0 width="584" height="279" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="9___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>9.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT>Solución:</div> <div> <IMG SRC="21fd44450.gif" border=0 width="580" height="69" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="10__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>10.</B></FONT> Solución:</div> <div> <IMG SRC="21fe23080.gif" border=0 width="547" height="264" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="11__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>11.</B></FONT> Solución:</div> <div> <IMG SRC="21fff3690.gif" border=0 width="499" height="105" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="12___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>12.</B></FONT><FONT SIZE="3" COLOR="#0000CC" FACE="Times New Roman" ><B> </B></FONT>Solución:</div> <div> <IMG SRC="22003dc50.gif" border=0 width="573" height="197" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="13___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>13.</B></FONT><FONT SIZE="3" COLOR="#0000CC" FACE="Times New Roman" ><B> </B></FONT>Solución:</div> <div> <IMG SRC="220108f00.gif" border=0 width="520" height="240" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="14___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>14.</B></FONT><FONT SIZE="3" COLOR="#0000FF" FACE="Times New Roman" ><B> </B></FONT>Solución:</div> <div> <IMG SRC="22026b430.gif" border=0 width="619" height="323" ALT="Documento Microsoft Office Word" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="15"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="220348e50.gif" border=0 width="584" height="741" ALT="Documento Microsoft Office Word" 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" 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