Here, when referring to potential energy, I will take gravitational potential energy as an example. Consider the following diagram where two point masses $m_1$ and $m_2$ at a distance $r$ from each other exert a gravitational force on each other. Here I used the vector form of the law of gravitation which can be referred to from here.
To find the gravitational potential energy, we need to find the work done by the gravitational force first. Here, I define the change potential energy as the negative work done by a conservative force and the gravitational potential energy as the negative work done by the gravitational force in bringing two objects from an infinite distance to a distance $r$. Since the gravitational force varies with distance, we need to integrate the forces from infinity to r. And hence, we get the expression for an infinitesimal change in gravitational potential energy as:
$dU = -dW_G$
$\implies dU = -\vec F_G\cdot\vec {dr}$
$\implies dU = -\frac{-Gm_1m_2}{r^2}\hat r_{1,2}\cdot\vec {dr}$
Since the two vectors are parallel, we can just take the product of their magnitudes as their dot product.
$\implies dU = \frac{Gm_1m_2}{r^2} dr$
On integrating and applying suitable limits, we get the expression for potential energy, the math can be seen here.
My question is, why do we take $\vec {dr}$ in the increasing direction of $r$? How can we assign an arbitrary direction to it? For example, when the particle moves from an infinite distance to a distance $r$, the displacement points towards the decreasing direction of $r$.
The explanations I received for this are that while integrating, the limits indicate the displacement of the particle and taking $dr$ in the opposite direction would mean that we are indicating the direction twice which leads to the wrong answer. But I was not satisfied with this explanation. It would be kind of you to clarify this doubt of mine. Thanks.
