Question

In the case of:

\int sin(x) dx

I am wondering if the dx is there denoting that sin(x) is a derivative or if the dx symbol in the case of the integral has something to do with the chain rule.

Answer 1

\int sin(x) dx is asking for the the value of a derivative of a function with respect to x that equals what's inside the integral.
int sin(x) dx means "what function, with respect to x, has a derivative of sin(x)?"

in this case, -cos(x) has a derivative of sin(x)... but, don't forget, all derivatives of constants are 0 (no slope - no change in 'y').

int sin(x) dx = -cos(x) + C.