Answer: The required laplace transform of g(t) is [tex]\dfrac{5}{(s+5)^2}.[/tex]
Step-by-step explanation: We are given to find the laplace transform of the following function :
[tex]g(t)=5te^{-5t}.[/tex]
We know the following formulas for laplace transform :
[tex](i)~L\{t^ne^{at}\}=\dfrac{n!}{(s-a)^{n+1}},\\\\(ii)~L\{cf(t)\}=cL\{f(t)\}.[/tex]
In the given function function, we have
c = 5, n = 1 and a = -5.
Therefore, we get
[tex]L\{g(t)\}\\\\=L\{5te^{-5t}\}\\\\=5L\{te^{-5t}\}\\\\\\=5\times\dfrac{1!}{(s-(-5))^{1+1}}\\\\\\=\dfrac{5}{(s+5)^2}.[/tex]
Thus, the required laplace transform of g is [tex]\dfrac{5}{(s+5)^2}.[/tex]
