From a table of integrals, we know that for $a, b \neq 0$, $$\int e^{at} \cos(bt) \,dt = e^{at} \cdot \displaystyle\frac{a\cos(bt) + b\sin(bt)}{a^2 + b^2} + C.$$

1. Assume $w$ is a constant, and use this antiderivative to compute the following improper integral:
   
   $\displaystyle \int_0^{\infty} \cos\!\left(wt\right) e^{-st} \,dt \ = \ \lim_{T\to\infty}$ if $s\neq 0$
   
   or
   
   $\displaystyle \int_0^{\infty} \cos\!\left(wt\right) e^{-st} \,dt \ = \ \lim_{T\to\infty}$ if $s=0.$ help (formulas)
   
   
2. For which values of $s$ do the limits above exist? In other words, what is the domain of the Laplace transform of $\cos\!\left(wt\right)$?
   
   help (inequalities)
   
   
3. Evaluate the existing limit to compute the Laplace transform of $\cos\!\left(wt\right)$ on the domain you determined in the previous part:
   
   $F(s) = {\mathcal L} \left\lbrace \cos\!\left(wt\right) \right\rbrace =$ help (formulas)