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Math Help - Laplace transform

  1. #1
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    Laplace transform

    What is the laplace transform of \displaystyle \int_0^t {\frac{\sin (t) \, dt}{t}} ?


    I got this from an old reviewer and its a multiple choice question. The selections are

    a. (arc tan s) / s
    b. (arc sin s) / s
    c. (arc cos s) / s
    d. (arc cot s) / s

    sorry for the typo
    Last edited by bibbonacci; December 22nd 2010 at 09:31 PM. Reason: typo
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bibbonacci View Post
    What is the laplace transform of \displaystyle \int_0^\infty {\frac{\sin t \, dt}{t}} ?
    Is this is typo? \displaystyle \mathcal{L}\left(\int_0^{\infty}\frac{\sin(t)}{t}\  text{ }dt\right)(s)=\frac{\pi}{2s}
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  3. #3
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    Drexel, thanks for noticing
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  4. #4
    A Plied Mathematician
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    Well, what do you get when you write out the definition of the LT as applied to your function?
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    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by bibbonacci View Post
    What is the laplace transform of \displaystyle \int_0^t {\frac{\sin (t) \, dt}{t}} ?


    I got this from an old reviewer and its a multiple choice question. The selections are

    a. (arc tan s) / s
    b. (arc sin s) / s
    c. (arc cos s) / s
    d. (arc cot s) / s

    sorry for the typo
    Starting from the L-transform of the sine function...

    \displaystyle \mathcal{L} \{\sin t\} = \frac{1}{1+s^{2}} (1)

    ...You first apply the 'division by t' rule...

    \displaystyle \mathcal{L} \{ \frac{f(t)}{t}\} = \int_{s}^{\infty} F(u)\ du \implies \mathcal{L} \{\frac{\sin t}{t} \} = \int_{s}^{\infty} \frac{du}{1+u^{2}} = \frac{\pi}{2} - \tan^{-1} s = \cot^{-1} s (2)

    ... and then the 'transform of integrals' rule...

    \displaystyle \mathcal{L} \{ \int_{0}^{t} f(\tau)\ d\tau\} = \frac{F(s)} {s} \implies \mathcal{L} \{\int_{0}^{t} \frac{\sin \tau}{\tau}\ d\tau \} = \frac{\cot^{-1} s}{s} (3)



    Merry Christmas from Italy

    \chi \sigma
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  6. #6
    A Plied Mathematician
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    I was going to go from the definition and use by parts or something like that.
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