Find the transfer function of the system given by its weight function
More often than not, one should not literally apply the definition of Laplace transform to compute the L-equivalent of a time domain function.

Let us just use linearity:
$$
\mathscr{L}[a\cdot f(t) + b\cdot g(t)]=a\cdot\mathscr{L}[f(t)]+b\cdot \mathscr{L}[g(t)]
$$
Where \$a,b\$ are scalar quantities.

Let's rewrite your \$w(t)\$:
$$
w(t)=
50\cdot(e^{-5t}-e^{-10t})\cdot u(t)=
50\cdot\Big(e^{-5t}\cdot u(t) - e^{-10t}\cdot u(t) \Big)
$$
Now we apply \$\mathscr{L}[\ \cdot\ ]\$ and use linearity:
$$
W(s)=\mathscr{L}[w(t)]=
50\Big[\mathscr{L}[e^{-5t}\cdot u(t)]-\mathscr{L}[e^{-10t}\cdot u(t)]\Big]
$$

Can you take it from here?