My years in graduate school instilled a Pavlovian response to PDEs: multiply by a test function and integrate by parts. This turns a differential equation into an integral equation .
I’ve been reading a book  on integral equations right now, and it includes several well-known techniques for turning certain kinds of integral equations into differential equations. Then this afternoon I talked to someone who was excited to have discovered a way to turn a more difficult integral equation into a differential equation.
For theoretical purposes, you often want to turn differential equations into integral equations. But for computational purposes, you often want to do the reverse.
Differential and integral equations are huge, overlapping fields, and sweeping generalities have exceptions. The opposite of the statement above may also be true. You may want to turn a differential equation into an integral equation for computational purposes (as in the finite element method) or turn an integral equation into a differential equation for theoretical convenience (as was the case for the person I was taking to).
 Sorta. More precisely this moves from the strong form of the PDE to a weak form. This involves integration, at least formally.
 Vladimir Ryzhov et al. Modern Methods in Mathematical Physics: Integral Equations in Wolfram Mathematica.