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June 17th, 2019, 01:49 PM   #1
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How to solve this simple Laplace integral?

Check this Image. I tried to post it as an image but it takes the entire space of the post...

It seems I have forgotten a lot about solving integrals... But I really need to remember quickly how to solve this kind of fractional integrals since I haven't memorized the Laplace "trick" formulas (like 1 over Laplace is $\displaystyle \frac{1}{s}$) and tomorrow I have a test.

Last edited by babaliaris; June 17th, 2019 at 01:53 PM.
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June 18th, 2019, 01:18 AM   #2
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I can't see the image for some reason. Can you just type enough to explain what the integral is?
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June 18th, 2019, 03:55 AM   #3
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Here...
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File Type: jpg vsfiPnN.jpg (94.4 KB, 7 views)
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June 18th, 2019, 05:37 AM   #4
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By the way I finished the test today, and it was 90% of Fourier transformation exercises while in the past you had like one exercise only... I probably failed the test and the weird thing is that i knew how to solve the integrals...

The problem is that the textbook says that $\displaystyle \mathcal{L}[1] = \frac{1}{s}$ but if I try to calculate it,

$\displaystyle
\mathcal{L}[1] = \lim_{T->\infty} \int_{-T}^{T} 1 \cdot e^{-st} dt =
\lim_{T->\infty} [\frac{e^{-st}}{-s}]_{-T}^{T} =
\lim_{T->\infty} [\frac{e^{-sT}}{-s} - \frac{e^{sT}}{-s}] =
\frac{1}{s} \lim_{T->\infty} [-e^{-sT} + e^{sT}]
$

I believe that $\displaystyle \lim_{T->\infty} [-e^{-sT}] = 0 $ and $\displaystyle \lim_{T->\infty} [e^{sT}] = \infty$ by trying to "see it" using the graph below.



But of course this is not true, because the limit of them both must be 1 in order to get $\displaystyle \frac{1}{s}$

This is a really great problem that is keeping me from calculating Laplace integrals in general.

I certainly need to review calculus all over again, but I have no time... I will try this summer though.... This limit is extremely easy and yet I don't remember how to solve it.

Offtopic: How did you put that image into a thumbnail? When I try to add a big image in a post usually it takes over the whole screen.

Last edited by babaliaris; June 18th, 2019 at 05:43 AM.
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June 18th, 2019, 11:14 AM   #5
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Quote:
Originally Posted by babaliaris View Post
How did you put that image into a thumbnail?
Post as an attachment instead of a URL.
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