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August 17th, 2012, 10:54 AM  #1 
Senior Member Joined: Aug 2012 From: New Delhi, India Posts: 157 Thanks: 0  Evaluation of (sin x)^m (cos x)^n dx
Given the values of m & n how would you evaluate without means of Gamma function? 
August 17th, 2012, 11:49 AM  #2 
Math Team Joined: Aug 2012 From: Sana'a , Yemen Posts: 1,177 Thanks: 44 Math Focus: Theory of analytic functions  Re: Evaluation of (sin x)^m (cos x)^n dx Just a Hint I would first rewrite is like this using integrating by parts I would continue until I get sin(x) 
August 17th, 2012, 01:41 PM  #3  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 407  Re: Evaluation of (sin x)^m (cos x)^n dx Hello, Rejjy! Quote: [color=beige]. . [/color] [color=beige]. . [/color] [color=beige]. . [/color] [color=beige]. . [/color] ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ [color=beige]. . . . . . [/color] [color=beige]. . . . . . [/color]  
August 18th, 2012, 01:19 AM  #4 
Senior Member Joined: Sep 2009 From: Wisconsin, USA Posts: 227 Thanks: 0  Re: Evaluation of (sin x)^m (cos x)^n dx
What about something along the lines of m=7/2 and n=3/2.

August 18th, 2012, 07:46 AM  #5 
Senior Member Joined: Aug 2012 From: New Delhi, India Posts: 157 Thanks: 0  Re: Evaluation of (sin x)^m (cos x)^n dx
And what if m and n are very large? I have a book that states the following result. Also, it says to multiple the above by if both m and n are even. @The_Fool Agreed, that is an interesting question. However, I am not aware if the above result holds for fractional m & n. It seems like it won't. Regards, Rejjy 18Aug2012 21:16 IST 
August 18th, 2012, 09:10 AM  #6  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Evaluation of (sin x)^m (cos x)^n dx Quote:
where is the complete elliptic integral of the first kind and is the gamma function. Balarka .  
August 18th, 2012, 09:22 AM  #7 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Evaluation of (sin x)^m (cos x)^n dx
Now for the original problem, I would prefer zaidalyafey's method : if then, where which is elementary and easy to calculate. Balarka . 
August 18th, 2012, 02:24 PM  #8  
Senior Member Joined: Sep 2009 From: Wisconsin, USA Posts: 227 Thanks: 0  Re: Evaluation of (sin x)^m (cos x)^n dx Quote:
 
August 18th, 2012, 10:22 PM  #9  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Evaluation of (sin x)^m (cos x)^n dx Quote:
 
August 19th, 2012, 01:16 AM  #10 
Senior Member Joined: Sep 2009 From: Wisconsin, USA Posts: 227 Thanks: 0  Re: Evaluation of (sin x)^m (cos x)^n dx
Well, I don't think that's right anyway since that integral is real: I used the beta function instead of the gamma function. While beta can be defined by gamma, it can stand on it's own without it in several other forms. 

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