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/(s^{2}+1)^{2 }
From the formula, L{f(t)} = ∫ e^{-st }t sin(t). How do I go about solving this integral?
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/(s^{2}+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 $\displaystyle \begin{align*} \int_0^{\infty}{e^{-s\,t}\,t\sin{(t)}\,dt} \end{align*}$.
If you don't want to use the fact that $\displaystyle sin(t)= \frac{e^{it}- e^{-it}}{2i}$ (which what romsek is suggesting) you can integrate by parts, taking $\displaystyle u= te^{-t}$ and $\displaystyle dt= sin(x)dt$.