may i know how to solve this question? i tried it , as seen in the picture, but got stuck
To be quite honest with you it was primarily because this is a famous example of a "central force" being conservative (gravitation), and I knew that in the one-dimensional case, it is quite obvious that this amounts to $\displaystyle \int -\frac{1}{r^3}\cdot r\, \frac{dr}{dt}\, dt=-\int\frac{1}{r^2}\,dr = \frac{1}{r}+C$.
The three-dimensional case is a little less obvious than this, but it also amounts to a simple substitution of $\displaystyle r := \sqrt{x^2+y^2+z^2}$, checking that $\displaystyle dt$ can be gotten rid of and then the integral really does turn out to have a value that is dependent on $\displaystyle r$ only, and therefore only on the location of the end-points of the path.
Moral of the story: Nothing beats memory when it comes to problem-solving . I am told that all experts mainly rely on memory (aka. experience), and use reasoning only when they absolutely have to.