1. Definite Integration

Hi,
Can anybody help me in integrating the following expression:

(1-x^n)^m

Integration is to be done with respect to x and the limits are from 0 to 1.

I could do the numerical integration or by expanding it binomially. Is there anyway of exact integration?

Thanks in anticipation

2. Originally Posted by dekar
Hi,
Can anybody help me in integrating the following expression:

(1-x^n)^m

Integration is to be done with respect to x and the limits are from 0 to 1.

I could do the numerical integration or by expanding it binomially. Is there anyway of exact integration?

Thanks in anticipation
Of course there is an exact formula if n,m are positive integers.

Because, like you said you can expand it by binomial to get a polynomial. And that polynomial can easily be expressed and its integral.
But it might be messy, I did not try.

Am I right to assume this is some problem you were given to find what the value is in terms of n and m? If thus, I might work on it if I have time.
The two recommendations I would say the is basic one, explain by binomial series.
The second is so use repeated use of integration by parts.

3. Hi and Thanks for your view on this problem.

m and n are positive real numbers, with specific small values of m and n (like (1,1), (2,1)) it is easier to solve. But to run it into a program I need an expression which contains the value of the integral in terms of m and n.

Thanks once again

Dekar

Originally Posted by ThePerfectHacker
Of course there is an exact formula if n,m are positive integers.

Because, like you said you can expand it by binomial to get a polynomial. And that polynomial can easily be expressed and its integral.
But it might be messy, I did not try.

Am I right to assume this is some problem you were given to find what the value is in terms of n and m? If thus, I might work on it if I have time.
The two recommendations I would say the is basic one, explain by binomial series.
The second is so use repeated use of integration by parts.

4. Originally Posted by dekar
Hi and Thanks for your view on this problem.

m and n are positive real numbers, with specific small values of m and n (like (1,1), (2,1)) it is easier to solve. But to run it into a program I need an expression which contains the value of the integral in terms of m and n.

Thanks once again

Dekar

try substitution u = x^n or x = u^(1/n)

it might reduce to beta integral and further to gamma function

beta and gamma functions

5. Originally Posted by qpmathelp
try substitution u = x^n or x = u^(1/n)

it might reduce to beta integral and further to gamma function

beta and gamma functions
The thing that I am afraid about that is the Gamma functions will produce irrational values usually. This problem is always rational.

I just graphed the function f(n,m) for n=m for the first ten values and tried to approximate it with different models. None of them satisfied me.

6. Hi

Thanks for suggestion.

This seems to be a useful substitution. I am getting the integral value as

(1/n)*beta((1/n), m+1), which can be calculated by the beta-gamma relationship.

I cant see the point which the Perfecthacker is trying to make.

Dekaar

7. Originally Posted by dekar
Hi and Thanks for your view on this problem.

m and n are positive real numbers, with specific small values of m and n (like (1,1), (2,1)) it is easier to solve. But to run it into a program I need an expression which contains the value of the integral in terms of m and n.

Thanks once again

Dekar
QuickMath gives what is in the attachment.

RonL