= Solution
Separate the constant embedding coordinate from its nonzero modes. Its integral gives the momentum-conservation delta function; below this is understood and absorbed into the amplitude normalization on the momentum-conserving locus. For the nonzero modes the Euclidean <Polyakov action> is quadratic, with <worldsheet Green function>
$$
\langle X^\mu(z)X^\nu(w)\rangle=-\frac{\alpha'}2\eta^{\mu\nu}\log|z-w|^2.
$$
Treat the <tachyon> vertex operators as normal ordered, removing coincident self-contractions. The <Gaussian functional integral>, or the same calculation by <Wick's theorem>, gives
$$
\left\langle\prod_r:e^{ik_r\cdot X(z_r)}:\right\rangle=\exp\left(-\sum_{r<s}k_r\cdot k_s\langle X(z_r)X(z_s)\rangle\right)=\prod_{r<s}|z_r-z_s|^{\alpha' k_r\cdot k_s}.
$$
All remaining determinants are independent of the insertion points and <momenta> and enter the overall constant. This derives the <Koba-Nielsen factor> and the requested position integral.
For the <Möbius transformation> $f(z)=(az+b)/(cz+d)$, $ad-bc=1$, one has
$$
f(z_r)-f(z_s)=\frac{z_r-z_s}{(cz_r+d)(cz_s+d)},\qquad d^2f(z_r)=|cz_r+d|^{-4}d^2z_r.
$$
The transformed <Koba-Nielsen factor> contributes, at insertion $r$, the power $-\alpha'\sum_{s\ne r}k_r\cdot k_s$ of $|cz_r+d|$. <Four-momentum conservation> makes this power $\alpha'k_r^2$. The complete differential integrand therefore transforms by
$$
\prod_r|cz_r+d|^{\alpha'k_r^2-4}.
$$
It is \b[invariant when $\alpha'k_r^2=4$ for every external <tachyon>]. The correlator alone is covariant; the integration measures are necessary for invariance. The <group> acting faithfully on the sphere is $PSL(2,\mathbb C)$, with the two elements $\pm I$ of $SL(2,\mathbb C)$ acting identically.
The apparent integral over all positions includes the residual conformal-group gauge volume. Divide by it and fix three distinct insertions. The <Faddeev-Popov determinant>, equivalently the <three-point worldsheet ghost correlator> and its conjugate, contributes
$$
J=|(z_1-z_2)(z_2-z_4)(z_4-z_1)|^2.
$$
Set $z_1=\Lambda$, $z_2=0$, $z_4=1$, then send $\Lambda$ to infinity on the sphere; the sign of the real direction is irrelevant. The factors involving $z_1$ behave as $|\Lambda|^{\alpha' k_1\cdot\sum_{s\ne1}k_s}=|\Lambda|^{-4}$, while $J\sim|\Lambda|^4$. They cancel, leaving $z_3=z$ as the one integration variable.
The remaining distance exponents are
$$
\alpha'k_2\cdot k_3=-4-\frac{\alpha't}{2},\qquad \alpha'k_3\cdot k_4=-4-\frac{\alpha's}{2}.
$$
For example $(k_2+k_3)^2=(k_1+k_4)^2=-t$ and each individual squared <momentum> is $4/\alpha'$. Also
$$
s+t+u=-\sum_{r=1}^4k_r^2=-\frac{16}{\alpha'}.
$$
Consequently the reduced amplitude is proportional to
$$
\int_{\mathbb C}d^2z\,|z|^{-4-\alpha't/2}|1-z|^{-4-\alpha's/2}.
$$
Apply the supplied complex beta integral with $A=4+\alpha't/2$, $B=4+\alpha's/2$, and $C=4-A-B=4+\alpha'u/2$. Its numerator arguments become $-1-\alpha't/4$, $-1-\alpha's/4$, $-1-\alpha'u/4$, while its denominator arguments become $2+\alpha'u/4$, $2+\alpha't/4$, $2+\alpha's/4$. Thus \b[the four-point amplitude is the <Virasoro–Shapiro amplitude>]
$$
\boxed{T_4=C_0\frac{\Gamma(-1-\alpha's/4)\Gamma(-1-\alpha't/4)\Gamma(-1-\alpha'u/4)}{\Gamma(2+\alpha's/4)\Gamma(2+\alpha't/4)\Gamma(2+\alpha'u/4)},}
$$
where $C_0$ is momentum-independent. The integral converges initially when $\operatorname{Re}A<2$, $\operatorname{Re}B<2$, and $\operatorname{Re}(A+B)>2$; the displayed answer elsewhere is its <analytic continuation>.
The <gamma function> has <simple poles> at nonpositive integers and no zeros. At generic fixed $t$, the amplitude therefore has \b[simple $s$-channel poles at $s=4(n-1)/\alpha'$, $n=0,1,2,\ldots$], with the same towers in $t$ and $u$ by <crossing symmetry>. They describe exchange of the <closed string> <tachyon>, massless states, and the infinitely many massive levels. The reciprocal denominator factors can give zeros at negative values $s=-4(\ell+2)/\alpha'$; special kinematics can also make an exchange <residue> vanish.
To see the structure of a generic <residue>, set $a=-1-\alpha's/4$, $b=-1-\alpha't/4$, $c=-1-\alpha'u/4$, so $a+b+c=1$. At $a=-n$, the <Gamma function recurrence> reduces the remaining factors to a constant times $\prod_{j=1}^n(b-j)^2$. The <Virasoro–Shapiro amplitude pole residue> is thus a polynomial of degree $2n$ in the other channel invariant, consistent with exchange up to spin $2n$. Crossed numerator poles must not simply be multiplied: near $a=-n+\delta$ and $b=-m+\varepsilon$, $c=1+n+m-\delta-\varepsilon$, and the zero of $1/\Gamma(1-c)$ cancels the putative double pole. The leading singular part is proportional to
$$
\binom{n+m}{n}^{\!2}\left(\frac1\delta+\frac1\varepsilon\right).
$$
The tree amplitude is <meromorphic>, with no loop threshold branch cuts; its channel singularities express factorization and string dual resonance.
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