The damped cosine function is given by,

x(t)=e−atcosωtu(t)=e−at(ejωt+e−jωt2)u(t)
By the definition of the Laplace transform, we have,

X(s)=L[e−atcosωtu(t)]=L[e−at(ejωt+e−jωt2)u(t)]
⇒L[e−atcosωtu(t)]=12{L[e−atejωtu(t)]+L[e−ate−jωtu(t)]}
⇒L[e−atcosωtu(t)]=12{L[e−(a−jω)tu(t)]+L[e−(a+jω)tu(t)]}
⇒L[e−atcosωtu(t)]=12[1s+(a−jω)+1s+(a+jω)]
⇒L[e−atcosωtu(t)]=12[1(s+a)−jω+1(s+a)+jω]=[s+a(s+a)2+ω2

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