How to evaluate the following (Laplace Transform)

Question

I know that the Laplace transform of a function $f(t)$ on $[0,\infty)$ is defined as \begin{equation} f^*(\theta)=\int^\infty_0e^{-\theta t}f(t) \, \mathrm{d}t. \end{equation}

But if I only have $f^*(\theta)=\dfrac{3+10\theta}{24\theta^2+22\theta+3}$, how to evaluate $f(t)$?

Answer 1

Use partial fractions to write

$$ f^*{\theta} = \frac{4}{7 (6 \theta +1)}+\frac{9}{7 (4 \theta +3)} $$

And now apply the fact that if $g(t)= e^{-a t}$, then

$$ g^*(\theta) = \frac{1}{\theta + a} $$

To find

$$ f(t) = \frac{1}{84} e^{-3 t/4} \left(8 e^{7 t/12}+27\right) $$