Results 1 to 5 of 5

Math Help - Integrability-Anti-Derrivatives Question

  1. #1
    Member
    Joined
    Jul 2009
    Posts
    168

    Integrability-Anti-Derrivatives Question

    Let f be a function that has an anti-derivative function F (F'(x)=f(x)) in [a,b].
    Prove / Disproof :

    f is integrable in [a,b].

    --

    Well, I said it shouldn't be, and I gave the example:
    <br />
F(x):= 0 , x=0
     x\sin\frac{1}{x} , 0<x\le1



    x\in[0,1]

    therefore,

    f:=\sin\frac{1}{x}-\frac{1}{x}cos\frac{1}{x} , x\in[0,1]

    f can't be integrable there, because it is not bounded for x\in(0,1].

    Is this a proper disproof?

    Than you
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,572
    Thanks
    1412
    Quote Originally Posted by adam63 View Post
    Let f be a function that has an anti-derivative function F (F'(x)=f(x)) in [a,b].
    Prove / Disproof :

    f is integrable in [a,b].

    --

    Well, I said it shouldn't be, and I gave the example:
    <br />
F(x):= 0 , x=0
     x\sin\frac{1}{x} , 0<x\le1



    x\in[0,1]

    therefore,

    f:=\sin\frac{1}{x}-\frac{1}{x}cos\frac{1}{x} , x\in[0,1]
    f(0)= \lim_{h\to 0}\frac{f(0+h)- f(0)}{h}= \lim_{h\to 0}\frac{hsin(1/h)}{h}= \lim_{h\to 0} sin(1/h) does not exist so this function is not defined on all of [0, 1].

    f can't be integrable there, because it is not bounded for x\in(0,1].

    Is this a proper disproof?

    Than you
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jul 2009
    Posts
    168
    Quote Originally Posted by HallsofIvy View Post
    f(0)= \lim_{h\to 0}\frac{f(0+h)- f(0)}{h}= \lim_{h\to 0}\frac{hsin(1/h)}{h}= \lim_{h\to 0} sin(1/h) does not exist so this function is not defined on all of [0, 1].
    That's right...

    Well, does that mean my disproof is wrong?

    Do I need to prove it?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Jul 2009
    Posts
    168
    I still don't understand - my original claim:

    Let f be a function that has an anti-derivative function F (F'(x)=f(x)) in [a,b].
    Prove / Disproof :

    f is integrable in [a,b].
    Is it true, or false?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Jul 2009
    Posts
    168
    Can anyone please help me with that? I need it urgently.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. integrability question
    Posted in the Calculus Forum
    Replies: 11
    Last Post: September 3rd 2011, 02:04 PM
  2. Replies: 1
    Last Post: January 16th 2010, 02:31 AM
  3. derrivatives of ln
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 17th 2008, 03:52 PM
  4. derrivatives
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 15th 2008, 12:41 PM
  5. Stupid, Easy Anti-Derivative Question.
    Posted in the Calculus Forum
    Replies: 12
    Last Post: August 29th 2007, 09:49 PM

Search Tags


/mathhelpforum @mathhelpforum