I have an exercise where I have to calculate the potential energy function $U(x)$ of this force $F$. I know the function is given by integrating $-F$, but how do I do this? $c$ is a constant, and a and $x$ are both variables, expressed in meters.
Do I just integrate $F$ in function of $x$, because they specifically ask the function $U(x)$?
Yes! If they ask you $U(x)$ you simply have to integrate in the variable $x$. So you can consider $a$ as a constant.
If they had asked you $U(x,a)$; you know that: $$\vec{F}(x,a)=-\vec{\nabla}U(x,a)$$ so you would have two variables in the integration.
Also, it wouldn't be so easy to evaluate the integral: $$\int sin\bigg(\frac{1}{a}\bigg)da=asin\bigg(\frac{1}{a}\bigg)-Ci\bigg(\frac{1}{a}\bigg)+const$$Where $Ci(x)$ is the cosine integral (a particular function). This is a solution I found using a calculator. So it's normal that they have asked you only $U(x)$.
