# Using Laplace transform to evaluate definite integrals?

I would like to know if there is some relation between Laplace Transform and similar definite integrals. For instance, if I know that

$$\mathcal{L}\{f(t)\}(s)=F(s),$$

have I some information about $$\int_a^be^{-st}f(t)dt$$?

It would be useful on functions defined by parts etc.

Many thanks!

• by definition $\mathcal{L}\left\{f(t)\right\}=\int_{0}^{\infty}e^{-st}f(t)dt=F(s)$ . If I understand, you want to know if you have $F(s)$, do I have information about $f(t)$. The answer is yes, it´s called the inverse transform, $\mathcal{L}^{-1}\left\{F(s)\right\}$. – Ricardo770 Aug 10 at 18:41
• @Ricardo770, many thanks! In fact, I have $f(t)$ and $F(s)$, but I have troubles to calculate $\int_a^be^{-st}f(t)dt$. So, I'd like to know if having $F(s)$ this integral has some property. Thank you so much. – Still_waters Aug 11 at 0:48
• You should look for the region of convergence of the laplace transform to see when its valid. – Ricardo770 Aug 11 at 1:28
• Say $F(s)$ exists for any $s\geq 0$. I have some information for instance to $\int_{5}^{100}e^{-st}f(t)dt$...? Many thanks! – Still_waters Aug 11 at 1:39
• It will depend on how well behaved $f(t)$ is. – Ricardo770 Aug 11 at 1:43

Do you know that $$\mathcal{L}(f(t)g(t)) = lim_{T \rightarrow \infty} \frac{1}{2\pi i} \int_{c - iT}^{c+iT}{F(\sigma)G(\tau - \sigma) d\sigma}$$ (where the integration is done along a line in the complex plane which is contained in the domain of convergence of $$F,G$$)? If you know $$F(s)$$ you could plug in $$g$$ as the indicator function $$1_{[a,b]}$$ which has a simple transform, assuming $$[a,b]\subseteq \mathbb{R}_{+}$$.
• Thats great, sounds like a $step$ in the right direction badum tss – Tom Ariel Aug 15 at 12:57