1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Binding an integral Rudin 6.14

  1. Nov 11, 2008 #1
    1. The problem statement, all variables and given/known data

    Let f(x) = integral [x to x+1] (sin(e^t)dt).

    Show that (e^x) * |f(x)| < 2

    and that (e^x) * f(x) = cos (e^x) - (e^-1)cos(e^(x+1)) + r(x) where:

    |r(x)| < Ce^-x, C is a constant

    2. Relevant equations

    integration by parts

    3. The attempt at a solution

    Well, this integral obviously isn't an elementary function but I don't really care what it's equal to, just that the product of its absolute value and e^x is less than 2 for all x>0.

    So I get:

    (x+1)sin(e^(x+1)) - xsin(e^x) - integral [x to x+1] (t(e^t)cos(e^t)dt)

    <= xsin(e^(x+1)) + sin(e^(x+1)) - xe^x + integral [x to x+1] ((te^t)dt) (since cos is between -1 and 1)

    = xsin(e^(x+1)) + sin(e^(x+1)) - xe^x + te^t [evaluated at x to x+1] - integral [x+1 to x] e^t.

    = xsin(e^(x+1)) + sin(e^(x+1)) - xe^x + (x+1)e^(x+1) - xe^x - e^(x+1) + e^x

    Which doesn't really get me any kind of maximum value on the function if I take the derivative of that function times e^x.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 11, 2008 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    But that's not true … e^1*|f(1)| > 7. :frown:

    Do you mean (e^-x) * |f(x)| < 2?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Binding an integral Rudin 6.14
  1. Rudin 8.5a (Replies: 6)

  2. Rudin 8.10 (Replies: 4)

  3. Rudin 8.6 (Replies: 8)

  4. Rudin 9.14 (Replies: 1)

  5. Rudin √2 (Replies: 1)

Loading...