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Math Help - Finding the Laplace using the definition

  1. #1
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    Finding the Laplace using the definition

    Ok, so I need to find the Laplace of f(t)=t sin(t) using the definition of Laplace. I already know that the answer is 2s/(s2+1)2

    From the formula, L{f(t)} = e-st​ t sin(t). How do I go about solving this integral?
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  2. #2
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    Re: Finding the Laplace using the definition

    Quote Originally Posted by Coeus View Post
    Ok, so I need to find the Laplace of f(t)=t sin(t) using the definition of Laplace. I already know that the answer is 2s/(s2+1)2

    From the formula, L{f(t)} = e-st​ t sin(t). How do I go about solving this integral?
    Probably the easiest way is to find the Laplace transform of $F(s)=\mathscr{L}\left\{ t e^{\imath t}\right\}$

    Then $\mathscr{L}\left\{t \sin(t)\right\} = \dfrac{F(s)-\overline{F(s)}} {2\imath}$
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    Re: Finding the Laplace using the definition

    Quote Originally Posted by Coeus View Post
    Ok, so I need to find the Laplace of f(t)=t sin(t) using the definition of Laplace. I already know that the answer is 2s/(s2+1)2

    From the formula, L{f(t)} = e-st​ t sin(t). How do I go about solving this integral?
    First of all, the definition is actually $\displaystyle \begin{align*} \mathcal{L} \left\{ f(t) \right\} = \int_0^{\infty}{e^{-s\,t}\,f(t)\,dt} \end{align*}$, so you are actually solving the DEFINITE integral $\displaystyle \begin{align*} \int_0^{\infty}{e^{-s\,t}\,t\sin{(t)}\,dt} \end{align*}$.
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    Re: Finding the Laplace using the definition

    If you don't want to use the fact that sin(t)= \frac{e^{it}- e^{-it}}{2i} (which what romsek is suggesting) you can integrate by parts, taking u= te^{-t} and dt= sin(x)dt.
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