I have an optimization problem. I understand the result of the first-order condition. What I don't understand is how the author arrives at equation 4.6. This is from Paul Krugman's book ("The Spatial Economy").
The problem: \begin{equation} min \int_{0}^{n} p(i)m(i) \,di \, \,\,\, s.t \,\,\,\,\left[ \int_{0}^{n} m(i)^{\rho} \,di \right]^{\frac{1}{\rho}} = M \tag{4.3} \end{equation}
Fist-order condition:
\begin{equation} \frac{m(i)^{\rho-1}}{m(j)^{\rho-1}}=\frac{p(i)}{p(j)} \,\,|\,\,i \neq j \tag{4.4} \end{equation}
solving equation 4.4 for m(i):
\begin{equation} m(i) = m(j) \left( \frac{p(i)}{p(j)} \right)^{\frac{1}{\rho-1}} \notag{} \end{equation}
replacing $m(i)$ in the constraint and solving for m(j):
\begin{equation} \left[ \int_{0}^{n} \left( \left( \frac{p(i)}{p(j)} \right)^{\frac{1}{\rho-1}} \right)^{\rho} \,di \right]^{\frac{1}{\rho}} = M \notag{} \end{equation}
\begin{equation} m(j) = \frac{p(j)^{\frac{1}{\rho-1}}}{\left[ \int_{0}^{n} p(i)^{\frac{\rho}{\rho-1}} \,di \right]^{\frac{1}{\rho}}}M \tag{4.5} \end{equation}
the problem is in the following equation. I don't know how the author arrived at this equation. He just says: "using (4.5) and integrating over all j gives"
\begin{equation} \int_{0}^{n} p(j)m(j)\,dj = \left[ \int_{0}^{n} p(i)^{\frac{\rho}{\rho-1}} \,di \right]^{\frac{\rho-1}{\rho}} \tag{4.6} \end{equation}
I would be grateful if someone could help me.
Multiply both sides of (4.5) by $p(j)$: $$p(j)m(j) = \frac{ p(j)p(j)^{\frac1{p-1}}}{C^{\frac{1}p}} M=\frac{p(j)^{\frac p{p-1}}}{C^{\frac{1}p}} M\tag1$$ where for brevity we write $$C:=\int_0^np(i)^{\frac p {p-1}}di.$$ Now integrate over $j$. Then (1) becomes $$\int_0^np(j)m(j)\,dj=\frac{\int_0^np(j)^{\frac p{p-1}}\,dj}{C^{\frac{1}p}} M= \frac{C}{C^{\frac{1}p}}M =C^{\frac{p-1}p}M=\left[\int_0^np(i)^{\frac p {p-1}}di\right]^{\frac{p-1}p}M. $$