  It is shown that the integrals of the Jacobi polynomials
\begin{equation*}%\label{eq:Fn^J}
  \int_0^t (t-\theta)^\delta P_n^{(\alpha-\frac12,\beta-\frac12)}(\cos \theta)
\left(\sin \tfrac{\theta}2\right)^{2 \alpha}
  \left(\cos \tfrac{\theta}2\right)^{2 \beta} d\theta > 0 \end{equation*} for
all $t \in (0,\pi]$ and $n \in \mathbb{N}$ if $\delta \ge \alpha + 1$ for
$\alpha,\beta \in \mathbb{N}_0$ and $\max\{\alpha,\beta\} > 0$. This proves a
conjecture on the integral of the Gegenbauer polynomials in \cite{BCX} that
implies the strictly positive definiteness of the function $\theta \mapsto (t -
\theta)_+^\delta$ on the unit sphere $\mathbb{S}^{d-1}$ for $\delta \ge \lceil
\frac{d}{2}\rceil$ and the Poly\`a criterion for positive definite functions on
the sphere for all dimensions. Moreover, the positive definiteness of the
function $\theta \mapsto (t - \theta)_+^\delta$ is also established on the
compact two-point homogeneous spaces.
