= Solution
Use the <Dirac delta function> to set $p'=p+\alpha T$, and write $X=x-x'$. The remaining momentum-dependent <quantum phase> is
$$
pX-\alpha Tx'-\frac{T}{2m}\left(p+\frac{\alpha T}{2}\right)^2
-\frac{\alpha^2T^3}{24m}.
$$
Shifting the integration variable by $\alpha T/2$ reduces the integral to the free <Gaussian integral>. Therefore
$$
\boxed{K_\alpha=\theta(T)\sqrt{\frac{m}{2\pi i\hbar T}}
\exp\left\{\frac i\hbar\left[
\frac{mX^2}{2T}-\frac{\alpha T(x+x')}{2}-\frac{\alpha^2T^3}{24m}
\right]\right\}}.
$$
The <force> is $-\alpha$, not $+\alpha$. Both the linear endpoint term and the cubic-time term disappear in the zero-force limit. The prefactor remains the free one.
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