Question

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

Answer 1

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 \[\int\limits\limits_{0}^{1}x^m(1-x)^n dx = \frac{m!n!}{(m+n+1)!}\]

Answer 2

http://books.google.com/books?id=wyIoOiBhPsAC&pg=RA1-PA262&lpg=RA1-PA262&dq=integrate+Beta(m,n)&source=bl&ots=vMLhQM5SKR&sig=4k1yvFIrs690kmwBjLmMydmxNQM&hl=en&sa=X&ei=n0H5T_n0EeT24QTf39DcBg&ved=0CFkQ6AEwBQ#v=onepage&q=integrate%20Beta(m%2Cn)&f=false

pages 262 and 263

Answer 3

But there is none about the function = \[ \frac{m!n!}{(m+n+1)!}\]

Answer 4

In my case x is to the power of m. It is different to beta function!

Answer 5

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!!

Answer 6

Really? Try me.

Answer 7

try me?????????

Answer 8

Is that the wrong expression? I'm not a native English speaker. What I meant is I am going to try it now.

Answer 9

which one is wrong????

Answer 10

Hmmn looks like a very messy steps. I wonder if there is faster one that looks like on page 264?

Answer 11

but i got the answer via way i told you!!

Answer 12

Wait I think I get it. It form a pattern.

Answer 13

ya the way is same as on 263 page!

Answer 14

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?

Answer 15

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. :(