Prove that by laplace transformation :

L{1}=1/s , s>0

Prove that

L{t^n} =n! /s^(n+1)

Prove that

L[e^at] =1/(s-a), s>a

Prove that

L{$Sin{at}$} =$\frac{a} {(s^{2}+a^{2})},s>0$

Prove that

L$Cosat=\frac{s} {s^{2}+a^{2}},s>0$

Prove that ​​​​​​

L{$Sinhat$}$=\frac{a} {s^{2}-a^{2}},s>|a|$​​

Prove that

L{$coshat$} =$\frac{s} {s^{2}-a^{2}},s>|a|$

Find the laplace transformation of

Sin2t cos3t

Find the laplace transformation of$Sin^{3}2t$

Find the laplace transformation of$Cosh^{3}2t$

Find the laplace transformation of$(\sqrt{t} +\frac{1}{\sqrt{t}})^{3}$