System Control Query - Find y(t) from Y(s)

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The discussion focuses on finding the inverse Laplace transform to derive y(t) from the given Y(s). The provided Y(s) is expressed as a fraction involving k and s, which is then transformed into a form suitable for inversion. The resulting expression for y(t) includes an integral that features an exponential decay and a sine function, indicating a damped oscillatory response. Participants are encouraged to share methods or insights on performing the inverse Laplace transform effectively. The thread emphasizes the importance of correctly manipulating the given function to achieve the desired time-domain representation.
DrMath
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Dear PF Friends,

Given the system to be controlled.
I'm stuck in finding the inverse laplace (to find y(t) from Y(S)) - any advice?
 

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Must forgive me if it not correct. I'm trying

Y(s)=\frac{1}{s} ( \frac{k}{s(s+1)+k}) = \frac{1}{s} ( \frac{k}{(s+1/2)^2+ (k-1/4)})

Invert
y(t)=\frac{k}{\sqrt{k-1/4}} \int_0^t e^{-t/2} \sin{(\sqrt{k-1/4})t } dt
 

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