Question

F(t)=sint×u(t-2) laplace transform

Answer 1

$F(t)=sint\times u(t-2)$

Let $(t-2)=z\implies dt=dz$

So $F(z)=sin(z+2)\times u(z)$

Laplace will be $F(s)$

As $F(s)=\int^{\infin}_{0}e^{-s(z+2)}sin(z+2)dz$

Using by parts integration,

$F(s)=sin(z+2)\frac{e^{-s(z+2)}}{-s}+\frac{1}{s}\int^{\infin}_{0}cos(z+2)e^{-s(z+2)}dz$

Apply by parts integration again, we get

$F(s)=\frac{e^{-2s}(s\times sin2+cos2)}{s^2+1}$