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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
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Re: Evaluation of (sin x)^m (cos x)^n dx

Hello, Rejjy!

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[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 18-Aug-2012 21:16 IST August 18th, 2012, 09:10 AM   #6
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Re: Evaluation of (sin x)^m (cos x)^n dx

Quote:
 Originally Posted by The_Fool What about something along the lines of m=7/2 and n=3/2.
If my calculations are correct (I'm pretty sure they are correct) :

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
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Re: Evaluation of (sin x)^m (cos x)^n dx

Quote:
Originally Posted by mathbalarka
Quote:
 Originally Posted by The_Fool What about something along the lines of m=7/2 and n=3/2.
If my calculations are correct (I'm pretty sure they are correct) :

where is the complete elliptic integral of the first kind and is the gamma function.

Balarka

.
But, you used the gamma function. The original post asked for a way to evaluate it without the gamma function. August 18th, 2012, 10:22 PM   #9
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Re: Evaluation of (sin x)^m (cos x)^n dx

Quote:
 Originally Posted by The_Fool But, you used the gamma function. The original post asked for a way to evaluate it without the gamma function.
It's impossible to do it without any type of special function for 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. Tags cos, evaluation, sin ,

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