Gravitational potential energy :

"The work done in bringing a body of unit mass from infinite distance to the given point in gravitational field of other body is called the gravitational potential energy at that point".

Suppose, the mass of earth is $M_{E}$ and radius $R_{E}$. We have to determine the gravitational potential energy of mass $m$ at point $P$ from distance $r$ from the centre of earth.

Suppose, at any instant body of mass $m$ lying at point $\mathrm{A}$ at a distance $x$ from the centre of earth $\mathrm{OP}=x$

The gravitational force on the body at this point,

$\mathrm{F}=\frac{\mathrm{GM}_{\mathrm{E}} m}{x^{2}}$

The work done to move the body to small displacement from the centre of earth $d \mathrm{~W}=\mathrm{F} d x$

$\therefore d \mathrm{~W}=\frac{\mathrm{GM}_{\mathrm{E}} m}{x^{2}} d x$

Total work done to bring the body from infinite $(D)$ distance to $x$ distance,

$\mathrm{W}=\int d \mathrm{~W}=\int_{\infty}^{r} \frac{\mathrm{GM}_{\mathrm{E}} m}{x^{2}} d x $

$\mathrm{~W}=\mathrm{GM}_{\mathrm{E}} m \int_{\infty}^{r} x^{-2} d x$

$\mathrm{~W}=\mathrm{GM}_{\mathrm{E}} m\left[-\frac{1}{x}\right]_{\infty}^{r}$