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\((\sqrt{t} +\frac{1}{\sqrt{t}})^{3} \)