= Solution
Use a mostly-plus target <Minkowski spacetime> metric and restore the <string tension> by $T=(2\pi\alpha')^{-1}$. After <Wick rotation>, the <Polyakov action> on a <Riemann surface> is
$$
S_E=\frac1{4\pi\alpha'}\int_\Sigma\sqrt h\,h^{\mu\nu}\partial_\mu X^a\partial_\nu X_a.
$$
The <Polyakov path integral> integrates over embeddings and <worldsheet metrics>, dividing by <worldsheet diffeomorphisms> and <Weyl transformations>. For a consistent flat <bosonic string theory>, take the <critical dimension of the bosonic string> $d=26$. <Conformal gauge> turns the embedding fields into free scalars; its <Faddeev-Popov determinant> is represented by the <bc system>. The remaining integrations are over <worldsheet moduli> and external insertion points.
A <tachyon vertex operator> is
$$
V_k(z,\bar z)={}:e^{ik\cdot X(z,\bar z)}:.
$$
The <worldsheet Green function> is $\langle X^a(z)X^b(w)\rangle=-(\alpha'/2)\eta^{ab}\log|z-w|^2$. The <operator product expansion> with the <holomorphic stress-energy tensor> gives <conformal weights> $(\alpha' k^2/4,\alpha' k^2/4)$. An <integrated string vertex operator> must have weights $(1,1)$, so the <mass-shell condition> is $k^2=4/\alpha'$, or tachyonic <mass> parameter $m^2=-4/\alpha'$.
At lowest order in the <string coupling>, the <worldsheet> is a <Riemann sphere>. For four external states, three insertion points can be fixed by a <Möbius transformation>. In the <worldsheet ghost field> description these three vertices carry $c\bar c$, while the fourth is integrated. The <three-point worldsheet ghost correlator> gives the squared product of the three fixed-point separations, precisely the <sphere gauge fixing for four string vertices> factor. Thus fixing three points is a gauge choice, rather than deleting three integrations without their Jacobian.
The <integral> over the constant embedding <worldsheet zero mode> gives $(2\pi)^d\delta^{(d)}(\sum_i k_i)$. The remaining <Gaussian integral>, using <normal ordering> to remove self-contractions, gives the <Koba-Nielsen factor>
$$
\left\langle\prod_{i=1}^4 V_{k_i}(z_i,\bar z_i)\right\rangle
\ \propto\
\delta^{(d)}\!\left(\sum_i k_i\right)
\prod_{i<j}|z_i-z_j|^{\alpha' k_i\cdot k_j}.
$$
Set $(z_1,z_2,z_3,z_4)=(0,z,1,\infty)$; the vertex at infinity is defined with its conformal-weight factor. Let all momenta be incoming and define
$$
s=-(k_1+k_2)^2,\qquad t=-(k_2+k_3)^2,\qquad u=-(k_1+k_3)^2.
$$
The <mass-shell condition> and <momentum conservation> imply $s+t+u=-16/\alpha'$. Since $\alpha'k_1\cdot k_2=-4-\alpha's/2$ and similarly for the pair $(2,3)$, the <sphere tachyon position integral> becomes
$$
\mathcal A_4^{(0)}
=\mathcal N g_s^2(2\pi)^d\delta^{(d)}\!\left(\sum_i k_i\right)
\int_{\mathbb C}d^2z\,|z|^{-4-\alpha's/2}|1-z|^{-4-\alpha't/2}.
$$
Here $\mathcal N$ depends on the normalization of external <string vertex operators>; it does not affect the kinematic dependence.
To evaluate the remaining <worldsheet modulus>, put $a=-1-\alpha's/4$ and $b=-1-\alpha't/4$. The <complex beta integral> is
$$
\int_{\mathbb C}d^2z\,|z|^{2a-2}|1-z|^{2b-2}
=\pi\frac{\Gamma(a)\Gamma(b)\Gamma(1-a-b)}
{\Gamma(1-a)\Gamma(1-b)\Gamma(a+b)}.
$$
For example, this identity follows by writing the two powers as Schwinger-parameter <gamma integrals>, doing the two-dimensional <Gaussian integral>, and changing the two positive parameters to their sum and ratio. The <complex beta integral> first holds for $\operatorname{Re}a,\operatorname{Re}b>0$ and $\operatorname{Re}(a+b)<1$; the scattering result is its <analytic continuation>. The third gamma-function argument is $1-a-b=-1-\alpha'u/4$. Absorbing $\pi$ into $\mathcal N$, the result is the \b[tree-level four-tachyon amplitude], the <Virasoro–Shapiro amplitude>:
$$
\boxed{\mathcal A_4^{(0)}
=\mathcal C g_s^2(2\pi)^d\delta^{(d)}\!\left(\sum_i k_i\right)
\prod_{q\in\{s,t,u\}}\frac{\Gamma(-1-\alpha'q/4)}{\Gamma(2+\alpha'q/4)}.}
$$
It is symmetric in the three <Mandelstam variables> and has generic poles at $q=4(N-1)/\alpha'$, matching the <bosonic string mass spectrum>. The formula is the leading term of <string perturbation theory>. Higher orders are constructed by the same <Polyakov path integral> prescription on higher-<genus> <worldsheets>, with four marked points and the corresponding ghost and <worldsheet moduli> measure; genus $g$ has coupling power $g_s^{2g+2}$.
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