The integral obviously doesn't depend on $p$, since it must be rotationally invariant, and, using polar coordinates centered on $p$, one obtains $$I(p,\alpha) = \mathrm{vol}(S^{n-1})\int_{-1}^1 e^{i\alpha t}(1-t^2)^{(n-2)/2}dtt}(1{-}t^2)^{(n-2)/2}dt =\frac{2\ \pi^{n/2}}{\Gamma(n/2)}\int_{-1}^1 \cos(\alpha t)(1{-}t^2)^{(n-2)/2}dt.$$ This latter integral can be evaluated by standard techniques, but, for . For odd values of $n$, it is expressed in Bessel functions, while, for even values of $n$ it is expressed in terms of elementary functions. Thus, for $n=2$, one has $$\int_{-1}^1 e^{i\alpha t}(1-t^2)^{(2-2)/2}dt \cos(\alpha t)(1{-}t^2)^{(2-2)/2}dt = \frac{2\sin\alpha}{\alpha},$$ while, for $n=3$, one has $$\int_{-1}^1 e^{i\alpha t}(1-t^2)^{(3-2)/2}dt \cos(\alpha t)(1{-}t^2)^{(3-2)/2}dt = \frac{\pi\ \mathrm{BesselJ}(1,\alpha)}{\alpha}, J_1(\alpha)}{\alpha},$$ where $J_1$ denotes the Bessel function of the first kind, and so on.
The integral obviously doesn't depend on $p$, since it must be rotationally invariant, and, using polar coordinates , you get that centered on $p$, one obtains $$I(p,\alpha) = \mathrm{vol}(S^{n-1})\int_{-1}^1 e^{i\alpha t}(1-t^2)^{(n-2)/2}dt.$$ This latter integral can be evaluated by standard techniques, but, for odd values of $n$, it is expressed in Bessel functions, while, for even values of $n$ it is expressed in terms of elementary functions. Thus, for $n=2$, one has $$\int_{-1}^1 e^{i\alpha t}(1-t^2)^{(2-2)/2}dt = \frac{2\sin\alpha}{\alpha},$$ while, for $n=3$, one has $$\int_{-1}^1 e^{i\alpha t}(1-t^2)^{(3-2)/2}dt = \frac{\pi\ \mathrm{BesselJ}(1,\alpha)}{\alpha},$$ and so on.
The integral obviously doesn't depend on $p$, since it must be rotationally invariant, and, using polar coordinates, you get that $$I(p,\alpha) = \mathrm{vol}(S^{n-1})\int_{-1}^1 e^{i\alpha t}(1-t^2)^{(n-2)/2}dt.$$ This latter integral can be evaluated by standard techniques, but, for odd values of $n$, it is expressed in Bessel functions, while, for even values of $n$ it is expressed in terms of elementary functions.