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Math Help - Laplace Tranform ?!?!

  1. #1
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    Laplace Tranform ?!?!

    The question is laplace transform of 4sin(at+b) where a and b are constants. The answer is is ((a)cos(b)+(s)sin(b))*4/(s^2+a^2)-
    i think table of integrals is used but how do they get to the answer.....

    Help would be greatly appreciated

    GAB
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  2. #2
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     \mathcal{L} [4 \sin(at+b)] = 4 \int_{0}^{\infty} e^{-st}\sin(at+b) \ dt

     = 4 \int_{0}^{\infty} e^{-st} \Big(\sin(b) \cos(at) + \cos(b) \sin(at)\Big) \ dt

     = 4 \sin(b) \int_{0}^{\infty} e^{-st} \cos(at) \ dt + 4 \cos(b) \int_{0}^{\infty} e^{-st} \sin(at) \ dt

     4 \sin(b) \frac {s}{s^{2}+a^{2}} + 4 \cos(b) \frac {a}{s^{2}+a^{2}}

     = \frac {4 \Big(a \cos(b) + s \sin(b) \Big) } {s^{2}+a^{2}}
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    Thanx tonnes fot that, it really really helped!!
    xxx
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by gabbee View Post
    The question is laplace transform of 4sin(at+b) where a and b are constants. The answer is is ((a)cos(b)+(s)sin(b))*4/(s^2+a^2)-
    i think table of integrals is used but how do they get to the answer.....

    Help would be greatly appreciated

    GAB
    The Laplace transform:

    [\mathcal{L}f](s)=\int_{-\infty}^{\infty} f(x) e^{-st}\ dt

    In this case:

    \mathcal{L}\left\{4 \sin(at+b)\right\}=4 \int_{-\infty}^{\infty} \sin(at+b) e^{-st}\ dt= <br />
=4\  \text{Im} \left[ \int_{-\infty}^{\infty} e^{-st}e^{i(at+b)} \  dt \right] = <br />
 4\  \text{Im} \left[e^{ib} \int_{-\infty}^{\infty} e^{(-s+ia)t} \ dt\right]

    and the rest is routine.

    CB
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