Answer
See below
Work Step by Step
Given: $F(s)=\frac{1}{s^2}\\
G(s)=\frac{e^{-\pi s}}{s^2+1}$
a) Using the Convolution Theorem
$L^{-1}[F(s)* G(s)]=L^{-1}[\frac{1}{s^2}.\frac{e^{-\pi s}}{s^2+1}]\\
=L^{-1}[\frac{1}{s^2}].L^{-1}[\frac{e^{-\pi s}}{s^2+1}]\\
=txu_{\pi}(t)\sin (t-\pi)\\
=\int^t_0 (t-x)u_{\pi}(x)\sin (x-\pi) dx\\
=\int^t_0 (t-x)u_{\pi}(x)\sin x dx\\
=u_{\pi}(t)\begin{bmatrix}
\sin x-x\cos x+t\cos x
\end{bmatrix}^t_{\pi}\\
=u_{\pi}(t)(\sin t-\pi + t)$
b) Using partial fractions.
$L^{-1}[F(s) * G(s)]=L^{-1}[\frac{1}{s^2}.\frac{e^{-\pi s}}{s^2+1}]\\
=L^{-1}[e^{-\pi s}(\frac{1}{s^2}-\frac{1}{s^2+1})\\
=L^{-1}[e^{-\pi s}\frac{1}{s^2}]-L^{-1}[e^{-\pi s}\frac{1}{s^2+1}]\\
=u_{\pi}(t)(t-\pi +\ sin t)$
