What is multidimensional fourier transform of $e^{-\mathrm{abs}(x)}$?

I know the answer for 1-dimension case. However, I cannot do integration for higher dimensional case.

I used spherical coordinate to arrive at the following, but I can't go further.

$\begin{align}\int_{\mathbb{R}^n}e^{-a |x| -i\xi x}dx &= \int_0^\infty dx \int_0^\pi d\theta \int d\Omega_{n-2} x^{n-1}\sin^{n-2}(\theta) e^{-ax-i|\xi|x \cos(\theta)} \\ &= C \int_0^\infty \int_0^\pi x^{n-1}\sin^{n-2}(\theta) e^{-ax-i|\xi|x \cos(\theta)}\end{align}$

where by $d\Omega_{n-2}$ I mean measure on $S^{n-2}$

For $n = 3$ case, it is simple, but I don't know how to do this integral for higher dimension.

What is multidimensional Fourier transform of $e^{-\mathrm{abs}(x)}$?

I know the answer for 1-dimension case. However, I cannot do integration for higher dimensional case.

I used spherical coordinate to arrive at the following, but I can't go further.

$\begin{align}\int_{\mathbb{R}^n}e^{-a |x| -i\xi x}dx &= \int_0^\infty dx \int_0^\pi d\theta \int d\Omega_{n-2} x^{n-1}\sin^{n-2}(\theta) e^{-ax-i|\xi|x \cos(\theta)} \\ &= C \int_0^\infty \int_0^\pi x^{n-1}\sin^{n-2}(\theta) e^{-ax-i|\xi|x \cos(\theta)}\end{align}$

where by $d\Omega_{n-2}$ I mean measure on $S^{n-2}$

For $n = 3$ case, it is simple, but I don't know how to do this integral for higher dimension.

I know the answer for 1-dimension case. However, I cannnot do integration for higher dimensional case.

I used spherical coordinate to arrive at the following, but I can't go further.

$\begin{align}\int_{\mathbb{R}^n}e^{-a |x| -i\xi x}dx &= \int_0^\infty dx \int_0^\pi d\theta \int d\Omega_{n-2} x^{n-1}\sin^{n-2}(\theta) e^{-ax-i|\xi|x \cos(\theta)} \\ &= C \int_0^\infty \int_0^\pi x^{n-1}\sin^{n-2}(\theta) e^{-ax-i|\xi|x \cos(\theta)}\end{align}$

where by $d\Omega_{n-2}$ I mean measure on $S^{n-2}$

For n = 3 case, it is simple, but I don't know how to do this integral for higher dimension.

What is multidimensional fourier transform of $e^{-\mathrm{abs}(x)}$?

I know the answer for 1-dimension case. However, I cannot do integration for higher dimensional case.

I used spherical coordinate to arrive at the following, but I can't go further.

$\begin{align}\int_{\mathbb{R}^n}e^{-a |x| -i\xi x}dx &= \int_0^\infty dx \int_0^\pi d\theta \int d\Omega_{n-2} x^{n-1}\sin^{n-2}(\theta) e^{-ax-i|\xi|x \cos(\theta)} \\ &= C \int_0^\infty \int_0^\pi x^{n-1}\sin^{n-2}(\theta) e^{-ax-i|\xi|x \cos(\theta)}\end{align}$

where by $d\Omega_{n-2}$ I mean measure on $S^{n-2}$

For $n = 3$ case, it is simple, but I don't know how to do this integral for higher dimension.