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Math Help - Laplace transform to calculate an integral.

  1. #1
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    Laplace transform to calculate an integral.

    Use Laplace transform (with respect to t)to calculate the integral



    I=\int([\cos(tx)/(x^2+a^2)]dx t\geqslant 0


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  2. #2
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    Quote Originally Posted by sublim25 View Post
    Use Laplace transform (with respect to t)to calculate the integral



    I=\int([\cos(tx)/(x^2+a^2)]dx t\geqslant 0

    [IMG]file:///C:/Users/Jamie/AppData/Local/Temp/msohtmlclip1/01/clip_image002.jpg[/IMG]

    So you have

    I =\int\frac{\cos(tx)}{x^2+a^2}dx

    but what are the limits of integration?
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    Limits are from 0 to infinity.
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    Let

    I(t) =\int_{0}^{\infty}\frac{\cos(tx)}{x^2+a^2}dx

    Then

    \mathcal{L}\{I\} =\int_{0}^{\infty}\frac{\mathcal{L}\{\cos(tx)\}}{x  ^2+a^2}dx=\int_{0}^{\infty}\frac{s}{s^2+x^2}\cdot \frac{1}{x^2+a^2}dx

    Now by partial factions we get

    \mathcal{L}\{I\} =\frac{s}{s^2-a^2}\int_{0}^{\infty}\left( \frac{1}{x^2+a^2}-\frac{1}{x^2+s^2}\right)dx=\frac{\pi}{2}\left( \frac{s}{a(s^2-a^2)}-\frac{1}{s^2-a^2}\right)

    Now if you take the inverse Laplace transform we get

    I=\frac{\pi}{2a}\cosh(at)-\frac{\pi}{2}\sinh(at)
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    Can you please explain how you get the partial fractions to come out to this?
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    Quote Originally Posted by sublim25 View Post
    Can you please explain how you get the partial fractions to come out to this?
    \frac{s}{(x^2+a^2)(x^2+s^2)}=\frac{Ax+B}{x^2+a^2}+  \frac{Cx+D}{x^2+s^2}

     s =(Ax+B)(x^2+s^2)+(Cx+D)(x^2+a^2)

    Now expand all of this out to get

    s=Ax^3+Bx^2+s^2Ax+s^2B + Cx^3+Dx^2+a^2Cx+a^2D

    s=(A+C)x^3+(B+D)x^2+(s^2A+a^2C)x+(s^2B+a^2D)

    So this gives 4 equations in the 4 unknowns A,B,C,D

    A+C=0 \quad B+D=0 \quad s^2A+a^2C=0 \quad s^2B+a^2D=s

    dont forget that s and a are constants. Now just solve this system.
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  7. #7
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    I am not sure what I am doing wrong, but when I solve the system, everything is cancelling out.
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  8. #8
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    Quote Originally Posted by sublim25 View Post
    I am not sure what I am doing wrong, but when I solve the system, everything is cancelling out.
    I don't know either. Please post what you have done. Note that both A and C are equal to 0.
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