= Solution
Inside the uniform sphere, $V_c^2=3\sigma^2r^2/r_0^2$. Thus, with $X=V_c/(\sqrt2\sigma)$, part (b) gives
$$
\frac{v_e^2}{2\sigma^2}=\frac92-X^2,\qquad 0\leq X\leq\sqrt{3/2}.
$$
The <Gaussian integral> needed for the <lowered Maxwellian velocity distribution> is
$$
\int_0^X y^2e^{-y^2}\,dy=\frac{\sqrt\pi}4\operatorname{erf}X-\frac X2e^{-X^2}.
$$
Substitution $v=\sqrt2\sigma y$ therefore yields
$$
\int_0^{V_c}f(v)v^2\,dv=n_0I(X),\qquad I(X)=\frac1{4\pi}\left[\operatorname{erf}X-\frac{2X}{\sqrt\pi}e^{-X^2}-\frac{4X^3}{3\sqrt\pi}e^{X^2-9/2}\right].
$$
Since $V_c<v_e$ throughout the core, the integrand is positive and so is $I(X)$ for $X>0$. Using $mn_0=\rho_0$, part (a)'s massive-test-body drag magnitude is
$$
D=-\dot V\big|_{\rm drag}=\frac{16\pi^2G^2M\rho_0\ln\Lambda}{V_c^2}I(X).
$$
The slow drift of a <circular orbit> is determined by <torque>, not by equating its changing <circular speed> to the tangential drag. Here $V_c=\omega r$, with constant $\omega=\sqrt{4\pi G\rho_0/3}$. The <specific angular momentum> $j=rV_c=\omega r^2$ obeys
$$
\dot j=2V_c\dot r=-rD,\qquad \dot r=-\frac{rD}{2V_c}<0.
$$
Hence the positive local decay timescale is
$$
\boxed{\tau_+\equiv\frac{r}{2|\dot r|}=\frac{V_c}{D}=\frac{V_c^3}{16\pi^2G^2M\rho_0\ln\Lambda\,I(X)}.}
$$
There is a genuine factor-of-two discrepancy with the requested denominator $32\pi^2$. That denominator would give $V_c/(2D)$, which is the timescale for a flat <galaxy rotation curve>, not this uniform-density core. It also follows from the incorrect replacement $\dot V_c=-D$, which drops the $V_c\dot r$ term in $d(rV_c)/dt$. The literal signed definition $r/(2\dot r)$ is negative for an inward drift; it equals $-\tau_+$. A positive decay time requires the absolute value or a minus sign.
This is the <slow circular inspiral under tangential drag> calculation. Its assumptions require $D$ small enough for nearly circular adjustment and a meaningful large <Coulomb logarithm in stellar dynamics>; the leading-log approximation is not uniform at $V_c\to0$. Also, integrating the truncated distribution all the way to $v_e$ does not give exactly $n_0$. The model uses $n_0$ as the stated core normalization; exact self-consistency of a sharply truncated uniform core with that distribution is not implied.
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