Topic: Evaluation of definite integral. I am able to prove the first part of the question by integration by parts' technique. However I am clueless on the second part of the question. http://dl.dropbox.com/u/63664351/Calculus%20-%20Definite%20integral%20-%20evaluation%20of%20definite%20integral.PNG
Here is the LATEX version: Show that \[\int\limits_{0}^{1}x^m(1-x)^n dx =\frac{n}{m+1}\int\limits\limits_{0}^{1}x^{m+1}(1-x)^{n-1}dx\] for \( m>0 \) and \( n>0 \). Hence, or otherwise, show that \[\int\limits\limits_{0}^{1}x^m(1-x)^n dx = \frac{m!n!}{(m+n+1)!}\]
But there is none about the function = \[ \frac{m!n!}{(m+n+1)!}\]
In my case x is to the power of m. It is different to beta function!
but if we take \[(1-x)^{n} \] as first function and x^m as second function and repeatedly integrate by parts,answer can be proved!!
Really? Try me.
try me?????????
Is that the wrong expression? I'm not a native English speaker. What I meant is I am going to try it now.
which one is wrong????
Hmmn looks like a very messy steps. I wonder if there is faster one that looks like on page 264?
but i got the answer via way i told you!!
Wait I think I get it. It form a pattern.
ya the way is same as on 263 page!
Hey, I am still wondering though. It took me a long time to determine which is u and which is dv. (When integrating by parts). Is there a quick way to determine which is u and dv in general case?
YEAAAHHHH!! GOT IT! Still confuse on how to determine which is u and dv in general though. I have to try it one by one. :(
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