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Homework Help: 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?
     
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