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June 17th, 2019, 01:49 PM  #1 
Senior Member Joined: Oct 2015 From: Greece Posts: 137 Thanks: 8  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. 
June 18th, 2019, 01:18 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,823 Thanks: 2160 
I can't see the image for some reason. Can you just type enough to explain what the integral is?

June 18th, 2019, 03:55 AM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond 
Here...

June 18th, 2019, 05:37 AM  #4 
Senior Member Joined: Oct 2015 From: Greece Posts: 137 Thanks: 8 
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. 
June 18th, 2019, 11:14 AM  #5 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond  

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