trig integrals | Clipmarks


	trig integrals
rumpus_204
4-11-2009 2:13 PM
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<div style="margin: 12px 0px; font-family: arial; color: #333333; background: #ffffff; border: solid 4px #e5e5e5; width: 100%; clear: left;"><div class="CM_CTB_Content_Wrap" style="margin: 0px; padding: 0px;background-color: #ffffff;"><div style="border-bottom: solid 1px #dcdcdc; white-space: nowrap; margin-bottom: 8px; background-color: #eeeeee ;background-image: url(http://clipmarks.com/images/source-bg.gif); background-repeat: repeat-x; height: 24px; line-height: 24px; vertical-align: middle; padding-bottom: 4px; color: #666666; font-size: 10px;" ><a href="http://clipmarks.com/clip-to-blog/" title="see clips that are hot right now"><img src="http://content.clipmarks.com/blog_embed/bb7f2352-171b-424d-82dc-db244a7c51c3/30F8DC1D-2C3A-4425-91D4-CEDF6AEABA19/" alt="" width="19" height="19" border="0" style="vertical-align: middle; margin: 0px 4px; display: inline; border: none; float:none;" /></a>clipped from <a title="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/TrigInt.html" href="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/TrigInt.html" style="font-size: 11px;">www.math.ucdavis.edu</a></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/TrigInt.html"><H3>INTEGRATION OF TRIGONOMETRIC INTEGRALS  </H3>

<P>
Recall the definitions of the trigonometric functions.
</P><UL>
   <UL>
      <UL>
<li style="margin-left:16px;padding-left: 0px;">	  
<IMG height="50" width="97" border="0" align="middle" alt="$ \displaystyle{ \tan x = { \sin x \over \cos x } } $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img1.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;">	  
<IMG height="49" width="94" border="0" align="middle" alt="$ \displaystyle{ \sec x = { 1 \over \cos x } } $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img2.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;">	  
<IMG height="49" width="155" border="0" align="middle" alt="$ \displaystyle{ \cot x = { \cos x \over \sin x } = { 1 \over \tan x } } $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img3.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;">	  
<IMG height="49" width="93" border="0" align="middle" alt="$ \displaystyle{ \csc x = { 1 \over \sin x } } $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img4.gif" />
</LI></UL>
     </UL>
</UL>
<P> 
The following indefinite integrals involve all of these well-known trigonometric functions.  Some of the following trigonometry identities may be needed.
</P><UL>
   <UL>
      <UL>
          <UL>
<li style="margin-left:16px;padding-left: 0px;">	A.)  
<IMG height="35" width="131" border="0" align="middle" alt="$ \cos^2 x + \sin^2 x = 1 $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img5.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;">
B.)  
<IMG height="15" width="142" border="0" align="bottom" alt="$ \sin 2x = 2 \sin x \cos x $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img6.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> C.)  
<IMG height="33" width="145" border="0" align="middle" alt="$ \cos 2x = 2 \cos^2 x - 1 $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img7.gif" />     so that      
<IMG height="49" width="138" border="0" align="middle" alt="$ \cos^2 x = \displaystyle{ 1+\cos 2x \over 2}$" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img8.gif" /> 
</LI><li style="margin-left:16px;padding-left: 0px;"> D.)  
<IMG height="35" width="143" border="0" align="middle" alt="$ \cos 2x = 1 - 2 \sin^2 x $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img9.gif" />     so that      
<IMG height="49" width="136" border="0" align="middle" alt="$ \sin^2 x = \displaystyle{ 1-\cos 2x \over 2} $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img10.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> E.)  
<IMG height="35" width="164" border="0" align="middle" alt="$ \cos 2x = \cos^2 x - \sin^2 x $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img11.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> F.)  
<IMG height="33" width="134" border="0" align="middle" alt="$ 1 + \tan^2 x = \sec^2 x $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img12.gif" />     so that     
<IMG height="33" width="134" border="0" align="middle" alt="$ \tan^2 x = \sec^2 x - 1 $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img13.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> G.)  
<IMG height="33" width="132" border="0" align="middle" alt="$ 1 + \cot^2 x = \csc^2 x $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img14.gif" />      so that      
<IMG height="33" width="132" border="0" align="middle" alt="$ \cot^2 x = \csc^2 x - 1 $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img15.gif" /> 
</LI></UL>
	</UL>
     </UL>
</UL>
<P>
It is assumed that you are familiar with the following rules of differentiation.
</P><UL>
   <UL>
      <UL>
<li style="margin-left:16px;padding-left: 0px;"> 
<IMG height="31" width="116" border="0" align="middle" alt="$ D (\sin x) = \cos x $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img16.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 
<IMG height="31" width="131" border="0" align="middle" alt="$ D (\cos x) = - \sin x $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img17.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 
<IMG height="33" width="125" border="0" align="middle" alt="$ D (\tan x) = \sec^2 x $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img18.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 
<IMG height="33" width="139" border="0" align="middle" alt="$ D (\cot x) = - \csc^2 x $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img19.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 
<IMG height="31" width="153" border="0" align="middle" alt="$ D (\sec x) = \sec x \tan x $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img20.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 
<IMG height="31" width="166" border="0" align="middle" alt="$ D (\csc x) = - \csc x \cot x $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img21.gif" />
</LI></UL>
     </UL>
</UL>
<P>
These lead directly to the following indefinite integrals.
</P><UL>
   <UL>
<li style="margin-left:16px;padding-left: 0px;"> 1.)   
<IMG height="51" width="176" border="0" align="middle" alt="$ \displaystyle{ \int \cos x \, \ dx } \ = \ \sin x + C $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img22.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 2.)   
<IMG height="51" width="191" border="0" align="middle" alt="$ \displaystyle{ \int \sin x \, \ dx } \ = \ - \cos x + C $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img23.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 3.)   
<IMG height="51" width="186" border="0" align="middle" alt="$ \displaystyle{ \int \sec^2 x \, \ dx } \ = \ \tan x + C $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img24.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 4.)   
<IMG height="51" width="199" border="0" align="middle" alt="$ \displaystyle{ \int \csc^2 x \, \ dx } \ = \ - \cot x + C $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img25.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 5.)   
<IMG height="51" width="214" border="0" align="middle" alt="$ \displaystyle{ \int \sec x \tan x \, \ dx } \ = \ \sec x + C $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img26.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 6.)   
<IMG height="51" width="227" border="0" align="middle" alt="$ \displaystyle{ \int \csc x \cot x \, \ dx } \ = \ - \csc x + C $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img27.gif" /> 
</LI></UL>
</UL>
<P>
The next four indefinite integrals result from trig identities and u-substitution.
</P><UL>
    <UL>
<li style="margin-left:16px;padding-left: 0px;">  7.)   
<IMG height="51" width="206" border="0" align="middle" alt="$ \displaystyle{ \int \tan x \, \ dx } \ = \ \ln \vert \sec x \vert + C $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img28.gif" /> 
</LI><li style="margin-left:16px;padding-left: 0px;">  8.)   
<IMG height="51" width="203" border="0" align="middle" alt="$ \displaystyle{ \int \cot x \, \ dx } \ = \ \ln \vert \sin x \vert + C $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img29.gif" /> 
</LI><li style="margin-left:16px;padding-left: 0px;">  9.)   
<IMG height="51" width="258" border="0" align="middle" alt="$ \displaystyle{ \int \sec x \, \ dx } \ = \ \ln \vert \sec x + \tan x\vert + C $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img30.gif" /> 
</LI><li style="margin-left:16px;padding-left: 0px;">  10.)   
<IMG height="51" width="256" border="0" align="middle" alt="$ \displaystyle{ \int \csc x \, \ dx } \ = \ \ln \vert \csc x - \cot x \vert + C $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img31.gif" />
</LI></UL>
</UL>
<P>
We will assume knowledge of the following well-known, basic indefinite integral formulas :
</P><UL>
<li style="margin-left:16px;padding-left: 0px;">   
<IMG height="55" width="154" border="0" align="middle" alt="$ \displaystyle{ \int x^n \,dx } = \displaystyle{ {x^{n+1} \over n+1 } + C } $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img32.gif" /> , where <IMG height="14" width="14" border="0" align="bottom" alt="$ n $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img33.gif" /> is a constant <IMG height="31" width="68" border="0" align="middle" alt="$ (n \ne -1 ) $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img34.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 
<IMG height="51" width="143" border="0" align="middle" alt="$ \displaystyle{ \int { 1 \over x } \,dx } = \displaystyle{ \ln \vert x\vert + C } $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img35.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 
<IMG height="51" width="127" border="0" align="middle" alt="$ \displaystyle{ \int e^x \,dx } = \displaystyle{ e^x + C } $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img36.gif" />
</LI><li style="margin-left:16px;padding-left: 0px;"> 
<IMG height="51" width="182" border="0" align="middle" alt="$ \displaystyle{ \int k f(x) \,dx } = k \displaystyle{ \int f(x) \,dx } $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img37.gif" /> , where <IMG height="15" width="13" border="0" align="bottom" alt="$ k $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img38.gif" /> is a constant
</LI><li style="margin-left:16px;padding-left: 0px;"> 
<IMG height="51" width="311" border="0" align="middle" alt="$ \displaystyle{ \int ( f(x) \pm g(x) ) \,dx } = \displaystyle{ \int f(x) \,dx } \pm \displaystyle{ \int g(x) \,dx } $" src="http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/trigintdirectory/img39.gif" /></LI></UL></blockquote></div><div style="margin: 0px 6px 6px 4px;"><table style="font-size: 11px;border-spacing: 0px;padding: 0px;" cellpadding="0" cellspacing="0" width="100%"><tr><td style="background:transparent;border-width:0px;padding:0px;">&nbsp;</td><td align="right" style="background:transparent;border-width:0px;padding:0px;width:107px" width="107"><a href="http://clipmarks.com/share/30F8DC1D-2C3A-4425-91D4-CEDF6AEABA19/blog/" title="blog or email this clip"><img src="http://content8.clipmarks.com/images/c2b-foot.png" border="0" alt="blog it" width="107" height="17" style="border-width:0px;padding:0px;margin:0px;" /></a></td></tr></table></div></div>
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