# How to solve this simple Laplace integral?

#### babaliaris

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.

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#### skipjack

Forum Staff
I can't see the image for some reason. Can you just type enough to explain what the integral is?

Forum Staff
Here...

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#### babaliaris

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.

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#### greg1313

Forum Staff
How did you put that image into a thumbnail?
Post as an attachment instead of a URL.

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