I'm reading the advanced macroeconomics by Romer in Chapter3.5 'The Romer Model'(4th edition).

I have no idea how the lagrangian made it like this.

L=\int_{i=0}^{A}p(i)L(i)di-\lambda\left\{[\int_{[i=0]}^{A}L(i)^{\phi} ]^{1/\phi}-1 ight\}

how to do first-order condition with respect to L(i) in integral function?

the answer is p(i)=\lambda\L(i)^{\phi-1}