Note that x does not depend on s and is therefore treated like a constant when we integrate our function with respect to s. So we then have

With x being a constant and the derivative of s being one we have

Finally

I would like to note that you thought it might equal

This assumes that x has a dependence on S, which it does not (we've only defined y in terms of s). But supposing that it did, say X WAS dependent on S. The above would still not be correct, we would need to use the product rule here

Clearly this is

Which is not eqaul to what you had.