In dalemberts wave eqn why is that we suddenly introduce a variable in the velocity as we integrate?

Question

in wave equation there is a step where we integrate .
f(x)+g(x)=integ(v(s)ds) from c to x .why is that s introduced? integ stands for the integration symbol the lower limit is c and the upper is x

Answer 1

The s is just a dummy variable for the integration. Since you are evaluating the integral from c to x, the result is simply a function of x.

Answer 2

in the damblerts equation when we integrate ant thing we have to introduce a small unknown thing suppose x to balance the equation as if integration is not the total but just assumed as the most accurate result so we have assumed a small quantity as s here to balance