If $\alpha_s\left(Q^2\right)=c / \log Q^2$, show that (10.37) leads to
$$
\frac{\int x^{n-1} q\left(x, Q^2\right) d x}{\int x^{n-1} q\left(x, Q_0^2\right) d x}=\left(\frac{\log Q^2}{\log Q_0^2}\right)^{A_n},
$$
where
$$
\begin{aligned}
A_n & =\frac{c}{2 \pi} \int_0^1 x^{n-1} P_{q q}(x) d x \\
& =\frac{c}{2 \pi} \frac{4}{3}\left(-\frac{1}{2}+\frac{1}{n(n+1)}-2 \sum_{j=2}^n \frac{1}{j}\right) .
\end{aligned}
$$
That is, in QCD, the moments ( $n \geq 1$ ) of the quark structure functions decrease as calculable powers of $\log Q^2 \cdot c$ is given by (7.65).