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For which value of a does int e^(ax) cos(x) dx converge?

  1. Dec 5, 2007 #1
    1. The problem statement, all variables and given/known data

    For which value the definite integral (0 to infinity) e^ax cos(x) dx converges? Calculate for the value of a.

    Mathieu
     
  2. jcsd
  3. Dec 5, 2007 #2
    You must show an attempt at a solution. Forum rules (and plus I'm lazy).
    Now, how do you evaluate an improper integral?

    Casey
     
  4. Dec 5, 2007 #3
    It depends if it's a type 1 or 2. For type 1 you must calculate the integral by replacing the infinity by a variable and then calculate the limit using this variable. For type 2 with a discontinuity you must split the integral in two parts and then calculate the integral using the same way than for type 1.

    Mathieu
     
  5. Dec 5, 2007 #4

    Gib Z

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    It is the first type. So replace the infinity with a variable, and evaluate the integral.
     
  6. Dec 5, 2007 #5
    ok the integral is:

    (((a cos(x))/((a^2)+1)) + (sin(x)/((a^2)+1))) e^ax

    but how do I calculate the limit?

    Mathieu
     
  7. Dec 5, 2007 #6

    Gib Z

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    I haven't calculated the indefinite integral myself, so I am not sure if yours is correct, but now your integral was the difference of that expression evaluated at b, as b goes to infinity, and that expression evaluated at 0.
     
  8. Dec 5, 2007 #7
    I understand the b to infinity part but I do not understand the expression evaluated at 0 part.

    Mathieu
     
  9. Dec 5, 2007 #8

    Gib Z

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    The Fundamental theorem of calculus states roughly that [tex]\int^b_a f(x) dx = F(b) - F(a)[/tex] where F'(x) = f(x). So that evaluated at 0 part corresponds to the -F(a) part.
     
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