Find the following

$(i)L[(cos2t-sin2t{)}^2]$

$L[\frac{e^{-3t}(1-cos2t)}{t}]$

$(iii)$

$L[H(t-)sin(t+)]$

$(iv)$

$Iff(t)={\displaystyle \underset{n=1}{\overset{k-1}{}}}(-1{)}^{n}H(t-3n),$ find $L[f(t)].$

$(v)Iff(t)={\displaystyle \underset{n=1}{\overset{k}{}}}[t-(n-2)],$ find $L[f(t)].$

State the Differentiation theorem.

Hence, find $L[t{e}^{-3t}cosh2t]$

Find $L[{\displaystyle \underset{0}{\overset{t}{}}}\frac{e^{3t}-1}{t}dt]$

Find $ifit$ exists:

$L[\frac{e^{-t}-e^{-3t}}{t}]$

Hence, evaluate the integral: ${\displaystyle \underset{0}{\overset{}{}}}\frac{e^{-t}-e^{-3t}}{t}dt$

$Iff(t)=\{\begin{array}{cc}-1,& 0t<mn>1</mn>\\ t-1,& t1\end{array}$

$(i)$ Represent $f(t)in$ terms $of$ unit step functions

$(ii)$ Find its Laplace transform