Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Laplace Transform Integral

  1. Jan 1, 2014 #1
    I posted this in the homework section, but I haven't received any help, so hopefully putting it in this section won't be an issue. I'm trying to compute the integral

    $$ \int_0^ \infty \frac{ \cos xt}{1 + t^2} \, dt $$

    using the Laplace transform. The first thing that catches my eye is the 1 /(1 + t^2) factor, which is equal to the Laplace transform of sin x:

    $$ = \int_0^\infty \cos xt \, L[ \sin x ] \, dt $$

    Any ideas?
  2. jcsd
  3. Jan 2, 2014 #2


    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    Why do you want to use the Laplace transformation here? I'd rather use the Fourier transformation. It's simpler to put it first in the exponential form. Your Integral is
    [tex]F(x)=\frac{1}{2}\int_0^{\infty} \frac{\exp(\mathrm{i} x t)+\exp(-\mathrm{i} x t)}{1+t^2}.[/tex]
    Substituting [itex]t'=-t[/itex] in the second integral you get after some algebra
    [tex]F(x)=\frac{1}{2} \int_{-\infty}^{\infty} \frac{\exp(\mathrm{i} x t)}{1+t^2}. \qquad (1)[/tex]
    Now we use the fact that
    [tex]\int_{-\infty}^{\infty} \frac{\mathrm{d} x}{2 \pi} \exp(-|x|) \exp(-\mathrm{i} t x)=\frac{1}{\pi (1+t^2)}.[/tex]
    Thus (1) is (up to a factor [itex]\pi/2[/itex]) the inverse of this Fourier transform. This gives
    [tex]F(x)=\frac{\pi \exp(-|x|)}{2}.[/tex]
  4. Jan 2, 2014 #3
    Ok, thanks for that. Do you also see any way to do it with the Laplace transform?
  5. Jan 3, 2014 #4


    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    Hm, I've no idea. Perhaps you can somehow use the convolution theorem with some clever trick?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Laplace Transform Integral
  1. Laplace transform (Replies: 1)

  2. LaPlace transform (Replies: 1)

  3. Laplace Transform of (Replies: 26)

  4. Laplace Transforms (Replies: 10)