Question

Find ${\int }_{x=\infty }^{x=R}\frac{GMm}{x^2}dx$ $(G,M$ and $m$ are constants)

• A. $\frac{GMm}{R}$
• B. $-\frac{GMm}{R}$
• C. $-\frac{GMm}{R^2}$
• D. $-\frac{GMm}{R^2}$

Answer 1

The correct option is B $-\frac{GMm}{R}$

${\int }_{x=\infty }^{x=R}\frac{GMm}{x^2}dx=GMm{\int }_{\infty }^R\frac{1}{x^2}dx$
$=GMm{\int }_{\infty }^R{x}^{-2}dx=GMm{\left[\frac{x^{-1}}{-1}\right]}_{\infty }^R$
$=-GMm{\left[\frac{1}{x}\right]}_{\infty }^R=-GMm[\frac{1}{R}-\frac{1}{\infty }]$
$=-\frac{GMm}{R}(\because \frac{1}{\infty }=0)$