# Thread: Finding the Laplace using the definition

1. ## Finding the Laplace using the definition

Ok, so I need to find the Laplace of f(t)=t sin(t) using the definition of Laplace. I already know that the answer is 2s/(s2+1)2

From the formula, L{f(t)} = e-st​ t sin(t). How do I go about solving this integral?

2. ## Re: Finding the Laplace using the definition

Originally Posted by Coeus
Ok, so I need to find the Laplace of f(t)=t sin(t) using the definition of Laplace. I already know that the answer is 2s/(s2+1)2

From the formula, L{f(t)} = e-st​ t sin(t). How do I go about solving this integral?
Probably the easiest way is to find the Laplace transform of $F(s)=\mathscr{L}\left\{ t e^{\imath t}\right\}$

Then $\mathscr{L}\left\{t \sin(t)\right\} = \dfrac{F(s)-\overline{F(s)}} {2\imath}$

3. ## Re: Finding the Laplace using the definition

Originally Posted by Coeus
Ok, so I need to find the Laplace of f(t)=t sin(t) using the definition of Laplace. I already know that the answer is 2s/(s2+1)2

From the formula, L{f(t)} = e-st​ t sin(t). How do I go about solving this integral?
First of all, the definition is actually \displaystyle \begin{align*} \mathcal{L} \left\{ f(t) \right\} = \int_0^{\infty}{e^{-s\,t}\,f(t)\,dt} \end{align*}, so you are actually solving the DEFINITE integral \displaystyle \begin{align*} \int_0^{\infty}{e^{-s\,t}\,t\sin{(t)}\,dt} \end{align*}.

4. ## Re: Finding the Laplace using the definition

If you don't want to use the fact that $sin(t)= \frac{e^{it}- e^{-it}}{2i}$ (which what romsek is suggesting) you can integrate by parts, taking $u= te^{-t}$ and $dt= sin(x)dt$.