Question:- Compute \(L\)\(({t^{2}e^{2t}+2\sin(t)\cos(t)+3})\). Show your steps.

Solution:- \(L\)\(({t^{2}e^{2t}+2\sin(t)\cos(t)+3})\)

\(=\)\(L(t^{2}e^{2t})\) +\(L\)\((2\sin(t)\cos(t))\) +\(L(3)\) ——————————-(\(1)\)

\(L(t^{2}e^{2t})\)\(=L(t^{2})\).\(L(e^{2t})\)

\(=\frac{2}{s^3}.\frac{1}{S-2}\), ——————————————————–(\(a\))

Since, \(L({t^n})\) \(=\) \(\frac{n!}{S^{n+1}}\)

\(\Rightarrow\)\(L(t^{2})\)\(=\) \(\frac{2}{S^{2+1}}\)

\(=\) \(\frac{2}{S^{3}}\)

Since, \(L(e^{at})\)\(=\)\(\frac{1}{(S-a)}\)\(=\)\(\frac{1}{S-2}\)

Now, \(L({2\sin(t)\cos(t)})\) Which can be written as

\(=L({\sin2(t)})\)

\(L(\sin2t)\)\(=\)\(\frac{2}{S^{2}+2^{2}}\)

\(=\)\(\frac{2}{(S^{2}+4)}\) ———————————————————–(\(b\))

Since, Laplace transformation of \(L\)\((\sin\)\(at\)) \(=\)\(\frac{a}{S^{2}+a^{2}}\)

now, \(L(3)\)\(=\frac{3}{S}\) ———————————————————-(\(c\))

Since, \(L(1)\)\(=\frac{1}{S}\)

Now putting the values of (\(a)\),

Now putting the values of (\(a)\),

Now putting the values of (\(a)\), (\(b)\) and (\(c)\) in equation