Correct Answer - Option 4 :
\(\frac{2s+1}{(s^2+s+1)^2}\)
Concept
Multiplication of function in one domain corresponds to the differentiation in another domain
If f(t) having the Laplace transform F(s) then t ⋅ f(t) will have the transform as
f(t) ↔\( - \frac{{dF\left( s \right)}}{{ds}}\)
Calculation:
Given function f(t) is . And g(t) = t ⋅ f(t)
Laplace transform of g(t) is:
\(L[g(t)] = - \frac{d}{{ds}}\left( {\frac{1}{{{s^2}s + 1}}} \right)\)
\( = - \frac{{\frac{{d\left( 1 \right)}}{{ds}}\left( {{s^2} + s + 1} \right) - 1 \times \frac{{d\left( {{S^2} + S + 1} \right)}}{{ds}}}}{{{{\left( {{S^2} + S + 1} \right)}^2}}}\)
\( = - \left( {\frac{{0 - 1\left( {2s + 1} \right)}}{{{{\left( {{s^2} + s + 1} \right)}^2}}}} \right)\)
\( = \frac{{2s + 1}}{{{{\left( {{s^2} + s + 1} \right)}^2}}}\)
