= Solution
A radial light ray in the <FLRW metric> travels <comoving radial distance> $d\chi=dt/a$ in units $c=1$. The <total comoving visibility radius> is therefore the full available <conformal time> interval,
$$
d_c=\int_0^\infty\frac{dt}{a(t)}=\int_0^\infty\frac{da}{a^2H(a)}.
$$
Insert the <Friedmann equation> from part (a), then put $y=1/a$ to get
$$
\boxed{d_c=\frac1{H_0}\int_0^\infty\frac{dy}{\sqrt{1-\Omega_M+\Omega_My^3}}.}
$$
For $0<\Omega_M<1$ the integrand is bounded at zero and decays as $y^{-3/2}$ at infinity, so the <total conformal lifetime of a flat matter-Lambda universe> is finite. Equivalently, this distance is the present <comoving particle horizon> plus the remaining comoving distance to the <cosmological event horizon>. This describes causal visibility under the assumed ability to see back to the initial singularity.
The substitution $u=y[\Omega_M/(1-\Omega_M)]^{1/3}$ also evaluates the integral using the <beta function>:
$$
d_c=\frac{\Omega_M^{-1/3}(1-\Omega_M)^{-1/6}}{3H_0}\,\mathrm B\!\left(\frac13,\frac16\right).
$$
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