Question

any1 mind to explain how to solve the following laplace transform using shifting theorem?

Answer 1

\[f(t)=f(t)[H(t-0)-H(t-7)]+f(t)H(t-7)\]\[f(t)=[H(t-0)-H(t-7)]+\cos(t)H(t-7)\]
for the 1st part, i can obtain the value from the table, \[L[H(t-0)]-L[H(t-7)]=\frac{1}{s}-\frac{e ^{-7s}}{s}\]but for the 2nd part, i expanded \[\cos(t)H(t-7)=H(t-7)\cos(t-7+7)\]\[=H(t-7)[\cos(t-7)\cos(7)-\sin(t-7)\sin(7)]\]\[=H(t-7)[\cos(t-7)\cos(7)]-H(t-7)[\sin(t-7)\sin(7)]\]how to apply the 2nd shifting theorem here? i'm very confused on the laplace transform of a heaviside function ==

Answer 2

\[L[H(t-a)f(t-a)]=e ^{-as}L[f(t)]\]but in the question, there's a constant cos(7) in there. can i just take it out?