In the solution to the problem below provided by my professor he applies a first order condition to the lagrangian for ct, getting pt for the partial derivative of the argument "$\frac{d}{dc_t(t)} \int p_t(w)*c_t(w)dw$". This is puzzling to me since the derivative of a indefinite integral of ct should be ct not 1.
What's going on with this derivative and is it correct?
$\frac{d}{dc_t(t)} \int p_t(w)*c_t(w)dw$ = $p_t(w)$
Edit: Removed hand written notes.