= Solution
Dropping <radiation in cosmology> in the <Friedmann equation> gives
$$
\tau_0\simeq H_0^{-1}\int_0^1\frac{da}{\sqrt{\Omega_m a+\Omega_\Lambda a^4}}.
$$
To evaluate this <matter-vacuum conformal age integral>, put $a=(\Omega_m/\Omega_\Lambda)^{1/3}y$. The square root in the denominator becomes $\Omega_m^{2/3}\Omega_\Lambda^{-1/6}\sqrt{y(1+y^3)}$, whereas $da=\Omega_m^{1/3}\Omega_\Lambda^{-1/3}dy$. Therefore
$$
\boxed{\tau_0\simeq H_0^{-1}\Omega_m^{-1/3}\Omega_\Lambda^{-1/6}
\int_0^{(\Omega_\Lambda/\Omega_m)^{1/3}}\frac{dy}{\sqrt{y(1+y^3)}}.}
$$
The radiation-free approximation is not valid arbitrarily close to the lower limit. Its extension to zero nevertheless gives a good leading <conformal time> estimate: the early-time correction has size $O(H_0^{-1}\sqrt{\Omega_r}/\Omega_m)$, small compared with the total age when <matter-radiation equality> occurs sufficiently early. The apparent $y^{-1/2}$ endpoint singularity is integrable.
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