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Thread: Improper Integral (Special Case?)

  1. #1
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    Improper Integral (Special Case?)

    Improper Integral (Special Case?)-20180221_185536-3-.jpg

    So, I only learned about improper integrals and how to find out if it converges or not by checking if the limit when c goes to infinity. I'm not sure what the problem is asking for in this case since it is put into 'context'. Can someone give me guidance to this problem? I'm not sure what the L symbol stands for either. And , im not sure what im suppose to find? Am i suppose to find the answer in terms of s?
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  2. #2
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    Re: Improper Integral (Special Case?)

    you must not have read the problem very carefully if you have to ask what $\mathscr{L}$ means.

    It's defined on the second line of the problem.

    Find that definition and then see if the problem makes more sense to you.
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  3. #3
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    Re: Improper Integral (Special Case?)

    As Romsek said, on the second line of the problem you are told that "L(f)" is the "Laplace transform" of f which is defined to be $L(f)= \int_0^\infty e^{-st}f(t) dt$ "s" is a constant while we are integrating with respect to t so the result will be a function of s.

    You are asked to find L(f) with f(t)= 5t. That is, you must integrate $\int_0^\infty e^{-st}(5t)dt= 5\int_0^\infty te^{-st}dt$. I recommend "integration by parts".

    For the second one $f(t)= e^{-2t}$. Integrate $\int_0^\infty e^{-2t}e^{-st}dt= \int_0^\infty e^{-(2+ s)t}dt$.

    For the last one $f(t)= sin(2t)$. Integrate $\int_0^\infty sin(2t)e^{-st}dt$. Again, I recommend "integration by parts". In fact, because the integrand, $f(t)e^{-st}$, is a product, "integration by parts" is often the best way to calculate the Laplace transform.
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    Re: Improper Integral (Special Case?)

    Oh, I di what you say which was integration by parts after i realized that f(t) is just 5t. I just wasn't sure what the final answer would be. I assume it would be in terms of s after taking the limit.
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