Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A conjecture about the rationallity of a definite integral

  1. Sep 1, 2011 #1
    Is it true that

    [itex] \int_0^1 f(x) dx \in \mathbb{Q} \Rightarrow \int_0^1 x f(x) dx \in \mathbb{Q} [/itex]

    ?

    (Suppose [itex] f(x) [/itex] integrable as needed)

    I thought of this conjecture yesterday and still couldn't prove it, I tried using integration by parts to relate it to the original, but didn't work.

    Any ideas? Or counterexamples?
    Thanks!
     
  2. jcsd
  3. Sep 1, 2011 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    It is false. Try
    [tex]f(x) = \pi x \sin(\pi x),\ xf(x) = \pi x^2 \sin(\pi x)[/tex]
     
  4. Sep 1, 2011 #3
    Wow, thanks!

    I couldn't find a counterexample, it seems it wasn't so trivial to find one (to me)

    How did you find it so fast?
     
  5. Sep 1, 2011 #4

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I just started with sin(πx) which gave 2/π so I changed it so πsin(πx). That gave 2. Then I figured multiplying by x would get π involved in the answer so I tried xπsin(πx) which gave 1, so I tried x2πsin(πx) which gave an irrational.
     
  6. Sep 1, 2011 #5
    Brilliant! Thank you.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: A conjecture about the rationallity of a definite integral
Loading...