# (a) Given, F (S) = {3 S^2 - 6 S + 9} / {(S + 1)(S - 2)^2}, find f (t). (b) Given, F(S) = {6 S^2 +...

## Question:

(a) Given, {eq}\displaystyle F (S) = \dfrac {3 S^2 - 6 S + 9} {(S + 1)(S - 2)^2} {/eq}, find {eq}f (t) {/eq}.

(b) Given, {eq}\displaystyle F(S) = \dfrac {6 S^2 + 2 S + 42} {(S - 1) (S^2 + 2)^2} {/eq}, find {eq}f (t) {/eq}.

## Laplace transform:

Laplace transform is an integral transform which transforms a function of a real variable t to a function of a complex variable s.

The Laplace transform of a function in the time domain {eq}f\left( t \right) {/eq} is given by

{eq}L\left\{ f\left( t \right) \right\} =F\left( s \right) =\int _{ 0 }^{ \infty }{ { e }^{ -st }f\left( t \right) dt } {/eq}

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(a) Given, {eq}\displaystyle F (S) = \dfrac {3 S^2 - 6 S + 9} {(S + 1)(S - 2)^2} {/eq}

Lets split the fuction into partial fractions

which gives,

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