Results 1 to 7 of 7

Math Help - Definite Integration

  1. #1
    Newbie
    Joined
    Mar 2007
    Posts
    12

    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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by dekar View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2007
    Posts
    12
    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











    Quote Originally Posted by ThePerfectHacker View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Jan 2007
    Posts
    42
    Quote Originally Posted by dekar View Post
    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
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by qpmathelp View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Mar 2007
    Posts
    12
    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
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by dekar View Post
    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
    Attached Thumbnails Attached Thumbnails Definite Integration-gash.jpg  
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 8
    Last Post: September 2nd 2010, 12:27 PM
  2. Definite integration help.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 7th 2009, 11:06 AM
  3. Definite Integration
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 4th 2009, 11:09 PM
  4. Integration, definite
    Posted in the Calculus Forum
    Replies: 0
    Last Post: October 6th 2008, 09:12 PM
  5. Definite Integration
    Posted in the Calculus Forum
    Replies: 0
    Last Post: April 9th 2007, 03:49 AM

Search Tags


/mathhelpforum @mathhelpforum