Find laplace transform of each following a) displaystyle{t}^{n} b) displaystyle cos{omega}{t} c) displaystyle sin{{h}}{left({c}{t}right)} d) displaystyle cos{{h}}{left({c}{t}right)}

Laplace transform
Find laplace transform of each following
a) $$\displaystyle{t}^{n}$$
b) $$\displaystyle \cos{\omega}{t}$$
c) $$\displaystyle \sin{{h}}{\left({c}{t}\right)}$$
d) $$\displaystyle \cos{{h}}{\left({c}{t}\right)}$$

2020-10-29
ANSWER FOR (A)We have to find Laplace transform of $$\displaystyle{t}^{n}$$, that is, $$\displaystyle{L}{\left\lbrace{t}^{n}\right\rbrace}$$.
Now from the Laplace Transform table we know that
$$\displaystyle{L}{\left\lbrace{t}^{n}\right\rbrace}=\frac{{{n}!}}{{{s}^{{{n}+{1}}}}}$$
This shows that
$$\displaystyle{L}{\left\lbrace{t}^{n}\right\rbrace}=\frac{{{n}!}}{{{s}^{{{n}+{1}}}}}$$
ANSWER FOR (B) We have to find Laplace transform of $$\displaystyle \cos{{\left(\omega{t}\right)}}$$, that is, L $$\displaystyle{\left\lbrace \cos{{\left(\omega{t}\right)}}\right\rbrace}$$
Now from the Laplace Transform table we know that
$$\displaystyle{L}{\left\lbrace \cos{{\left({a}{t}\right)}}\right\rbrace}=\frac{s}{{{s}^{2}+{a}^{2}}}$$
Whis shows that
$$\displaystyle{L}{\left\lbrace \cos{{\left(\omega{t}\right)}}\right\rbrace}=\frac{s}{{{s}^{2}+\omega^{2}}}$$
ANSWER FOR (C) We have to find Laplace transform of $$\displaystyle \sin{{h}}{\left({c}{t}\right)}$$, that is, L $$\displaystyle{\left\lbrace \sin{{h}}{\left({c}{t}\right)}\right\rbrace}$$
Now from the Laplace Transform table we know that
$$\displaystyle{L}{\left\lbrace \sin{{h}}{\left({a}{t}\right)}\right\rbrace}=\frac{a}{{{s}^{2}-{a}^{2}}}$$
Whis shows that
$$\displaystyle{L}{\left\lbrace \sinh{{\left({c}{t}\right)}}\right\rbrace}=\frac{c}{{{s}^{2}-{c}^{2}}}$$
ANSWER FOR (D) We have to find Laplace transform of $$\displaystyle \cos{{h}}{\left({c}{t}\right)}$$, that is, L $$\displaystyle{\left\lbrace \cos{{h}}{\left({c}{t}\right)}\right\rbrace}$$
Now from the Laplace Transform table we know that $$\displaystyle{L}{\left\lbrace \cos{{h}}{\left({a}{t}\right)}\right\rbrace}=\frac{s}{{{s}^{2}-{a}^{2}}}$$
Whis shows that
$$\displaystyle{L}{\left\lbrace \cos{{h}}{\left({c}{t}\right)}\right\rbrace}=\frac{s}{{{s}^{2}-{c}^{2}}}$$