Suppose \(a>0\). The continuous integrand is at most one on
\([-1,1]\). For \(|x|\geq 1\), it is at most \(e^{-a|x|/2}\), whose
integral on \(\mathbb R\) is \(4/a\). This proves finiteness before
evaluation. On \([-1,1]\) the integrand is at least \(e^{-a/2}\), giving
the strictly positive lower bound \(2e^{-a/2}\).
Set \(J=\int _0^\infty e^{-x^2}\,dx\), likewise finite and positive.
For each fixed \(y>0\), the substitution \(x=yt\) gives
\(J=\int _0^\infty y e^{-y^2t^2}\,dt\). Multiply by \(e^{-y^2}\),
integrate in \(y\), and apply nonnegative Tonelli:
\[\begin {aligned}
J^2&=\int _0^\infty \int _0^\infty
y e^{-(1+t^2)y^2}\,dy\,dt\\
&=\int _0^\infty \frac {dt}{2(1+t^2)}=\frac \pi 4.
\end {aligned}\]
The inner primitive is \(-e^{-(1+t^2)y^2}/(2(1+t^2))\); the final
integral is evaluated by arctangent. Thus \(J=\sqrt {\pi }/2\).
Reflection and the positive substitution \(x=\sqrt {2/a}\,z\) give
the asserted value of \(I(a)\). If \(a\leq 0\), the integrand is at
least one everywhere and its nonnegative integral is infinite.