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Math Help - Laplace Transform

  1. #1
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    Laplace Transform

    This problem states using integration by parts to find the Laplace transform of the given function; n is a positive integer and a is a real constant:

    <br />
t^2 sin at<br />
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  2. #2
    MHF Contributor red_dog's Avatar
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    \displaystyle\int t^2\sin atdt=\int t^2\left(-\frac{\cos at}{a}\right)'dt=
    \displaystyle =-\frac{1}{a}t^2\cos at+\frac{2}{a}\int t\cos atdt=
    \displaystyle =-\frac{1}{a}t^2\cos at+\frac{2}{a}\int t\left(\frac{\sin at}{a}\right)'dt=
    \displaystyle =-\frac{1}{a}t^2\cos at+\frac{2}{a^2}t\sin at-\frac{2}{a^2}\int\sin atdt=
    \displaystyle =-\frac{1}{a}t^2\cos at+\frac{2}{a^2}t\sin at+\frac{2}{a^3}\cos at+C
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by red_dog View Post
    \displaystyle\int t^2\sin atdt=\int t^2\left(-\frac{\cos at}{a}\right)'dt=
    \displaystyle =-\frac{1}{a}t^2\cos at+\frac{2}{a}\int t\cos atdt=
    \displaystyle =-\frac{1}{a}t^2\cos at+\frac{2}{a}\int t\left(\frac{\sin at}{a}\right)'dt=
    \displaystyle =-\frac{1}{a}t^2\cos at+\frac{2}{a^2}t\sin at-\frac{2}{a^2}\int\sin atdt=
    \displaystyle =-\frac{1}{a}t^2\cos at+\frac{2}{a^2}t\sin at+\frac{2}{a^3}\cos at+C
    He is asking for the Laplace transform of t^2 \sin(at) so the
    integral in question is:

    <br />
\int_0^{\infty} t^2 \sin(at) e^{-st}dt<br />

    RonL
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