August 23rd, 2012, 12:07 PM  #1 
Math Team Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 Math Focus: Theory of analytic functions  Evaluate the integral 
August 23rd, 2012, 02:30 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,544 Thanks: 1476 
What have you tried?

August 24th, 2012, 11:05 AM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,720 Thanks: 984 Math Focus: Elementary mathematics and beyond  Re: Evaluate the integral 
August 25th, 2012, 08:10 AM  #4 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Evaluate the integral
Another way to do that is to immediately use to write the integral as Since that is an odd power of cosine, we can multiply both numerator and denominator by cos(x): Now let u= sin(x) so that du= cos(x)dx and the integral is which can be integrated by "partial fractions". 
August 25th, 2012, 12:16 PM  #5 
Senior Member Joined: May 2011 Posts: 501 Thanks: 5  Re: Evaluate the integral
May I ask something that has nothing to do with the integral at hand?. zaidalyafey, from where did you get the series in your avatar?. This appears to be the sum of the reciprocals of the Fibonacci numbers minus the first few terms. This does not converge to the gamma constant. 
August 26th, 2012, 06:55 AM  #6  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Evaluate the integral Quote:
http://www.mymathforum.com/viewtopic.php?f=15&t=32679  
August 26th, 2012, 08:46 AM  #7 
Senior Member Joined: May 2011 Posts: 501 Thanks: 5  Re: Evaluate the integral
Ok, mathbalarka. Often times, is used with reference to the sum of the reciprocals of the Fibonacci numbers. Which is irrational and equal to around 3.359.... or something like that. I do not like this notation either, as is used for the digamma. 

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