I am trying to evaluate $\int_{0}^{\infty} t \sin (t) dt$ by using Laplace transformation but I am getting inconsistent results.
Here is my solution:
$I=\int_{0}^{\infty} t \sin (t) dt$
$J(s):=\int_{0}^{\infty} t \sin (t)e^{-st} dt \implies J(0) =I$
Since $J(s)$ is the Laplace transform of $t\sin(t)$, therefore it is equal to $\dfrac{2s}{(s^2+1)^2} \implies J(s)=\dfrac{2s}{(s^2+1)^2}\implies J(0) =0$
On the other hand the integral of $t\sin(t)$ is $\sin(t)-t\cos(t)$ whose value keeps oscilating, so the integral should be undefined rather $0$.
What went wrong with the laplace transformation and how can I avoid it in future?
PS: Since I am not familiar with this forum, I am not sure what tags to add. I would be grateful if somebody could do it for me.