= Solution
Normalize the <scale factor> by $x=a/a_0$. <Constant-equation-of-state density scaling> for <pressureless matter> gives $\rho=\rho_0x^{-3}$. Multiplying the <Friedmann equation> by $x^2$ and substituting the expressions from part (i),
$$
\dot x^2=\frac{8\pi G\rho_0}{3x}-\frac{kc^2}{a_0^2}=H_0^2\left(\frac{2q_0}{x}+1-2q_0\right).
$$
Take the expanding branch $\dot x>0$ and set the origin of <cosmic time> at the <Big Bang>, $x=0$. The <age of a dust universe without a cosmological constant> is consequently
$$
\boxed{t_0=\frac1{H_0}\int_0^1\left(1-2q_0+\frac{2q_0}{x}\right)^{-1/2}\,dx.}
$$
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