Find Laplace Transform of each following (a) tn , (b) cosωt

For A
We have to find Laplace transform of ${\displaystyle t^n}$ , that is ${\displaystyle L\left\{t^n\right\}}$
Now from the Laplace Transform table
${\displaystyle L\left\{t^n\right\}=\frac{n!}{s^{n+1}}}$
It shows that:
${\displaystyle L\left\{t^n\right\}=\frac{n!}{s^{n+1}}}$
For B
We have to find Laplace transform of $cos\omega t$. That is $Lcos\omega t$

 

From the Laplace Table:

$L\cos at=s/(s^2+a^2)$

$L\cos \omega t=s/(s^2+{\omega }^2)$

 

For C

We have to find Laplace transorm of $\sinh (ct)$ that is $L\sinh ct$

 

From the Laplace Table:

$L\sinh at=a/(s^2+a^2)$

 

It shows that:

$L\sinh ct=c/(s^2+c^2)$

 

For D

We have to find Laplace transorm of $\cosh (ct)$that is $L\cosh ct$

 

From the Laplace Table:

$L\cosh at=s/(s^2+a^2)$

 

It shows that:

$L\cosh ct=s/(s^2+c^2)$