Integration Under the integral sign on indefinite integrals

Question

Is it possible to perform the differentiation under the integral sign for an indefinite integral (anti-derivative)? that is,

if $f(s) = \int F(s,t) dt $

then, is

$f'(s) = \int (d/ds(F(s,t)))dt$

where all the integrations are indefinite (without any limits)?

Answer 1

It doesn't make much sense, because $\int F(s,t) dt $ is not a function of $s$ as it has infinitely many values for each $s$.

Answer 2

For nicely behaved functions you can do this $$ \frac{\partial}{\partial s}\int F(s,t)dt = \int \frac{\partial}{\partial s} F(s,t) dt $$ If $s,t$ are completely independent. I know I used this trick when trying to combine coupled pdes in to a governing pde.