When an integral is taken with respect to one variable, any other independent symbols are treated as constants during that integration.

For example, integrating

$$xy+y^2$$

with respect to $x$ gives

$$\int(xy+y^2),dx =\frac{x^2y}{2}+xy^2+C(y).$$

Why the constant may depend on a parameter

Differentiating with respect to $x$ removes any term that depends only on $y$:

$$\frac{\partial}{\partial x}C(y)=0.$$

So the most general antiderivative with respect to $x$ contains an arbitrary function of the variables being treated as parameters, not merely one numerical constant.

Changing the integration variable

If the same expression is integrated with respect to $y$ instead,

$$\int(xy+y^2),dy =\frac{xy^2}{2}+\frac{y^3}{3}+C(x).$$

The notation $dx$ or $dy$ therefore matters: it determines which variable changes and which variables remain fixed.

This parameter viewpoint is the basic step behind iterated multiple integrals, where one variable is integrated at a time.