definite Integral

juantheron · Bericht door **juantheron** » 04 mar 2013, 07:37

Calculate value of $\int_{0}^{1}x^{30}.(1-x)^{70}dx =$

How can I calculate without using Gamma function

Thanks.

Bericht door **op=op** » 04 mar 2013, 10:37

partial integration, 70 times.
That seems to be a lot of work, but there is a nice pattern.

juantheron · Bericht door **juantheron** » 09 mar 2013, 10:00

Thanks op=op Got it

$\int_{0}^{1}x^m.(1-x)^n dx = \frac{m!.n!}{(m+n+1)!}$

Proof:: Let $I_{m,n} = \int (1-x)^n . x^m dx$

$= \frac{n}{m+1}\int (1-x)^{n-1}.x^{m+1}dx = \frac{n}{m+1}.I_{n-1,m+1}$

$= \frac{(n).(n-1)}{(m+1).(m+2)}I_{n-2,m+2}$

$= \frac{(n).(n-1).................1}{(m+1).(m+2)..............(m+n)}I_{0,m+n}$

So $I_{m,n} = \int_{0}^{1}x^m.(1-x)^ndx = \frac{n!.m!}{(m+n)!}.\frac{1}{m+n+1} = \frac{m!.n!}{(m+n+1)!}$

bcz $I_{m+n,0} = \int_{0}^{1}x^{m+n}dx = \frac{1}{m+n+1}$

So $I_{30,70} = \int_{0}^{1}x^{30}.(1-x)^{70}dx = \frac{30! . 70!}{101!}$

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