Find -x-x-2 RG M m-x2 d xG- M and m are constants GM mC- - GMm - R 2D- GM m 2- R

The correct option is A GMm2R nbsp;∫^x = 2 R_x = ∞GMmx^2 dx = GMm ∫_∞^2R1x^2dx = GMm ∫_∞^2R x^ 2 dx = GMm [ x^ 1 1 ]^2R_∞ = GMm [ 1x ]^2R_∞ = GMm [ 12R ...

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Standard XII

Physics

Definite Integrals

Find ∫x=∞x=2 ...

Question

Find $\int_{x = \infty}^{x = 2 R} \frac{G M m}{x^{2}} d x$ $(G, M$ and $m$ are constants)

\frac{G M m}{R}

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- \frac{G M m}{2 R}

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- \frac{G M m}{R}

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- \frac{G M m}{R^{2}}

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Solution

The correct option is B $- \frac{G M m}{2 R}$
$\int_{x = \infty}^{x = 2 R} \frac{G M m}{x^{2}} d x = G M m \int_{\infty}^{2 R} \frac{1}{x^{2}} d x$
$= G M m \int_{\infty}^{2 R} x^{- 2} d x = G M m {[\frac{x^{- 1}}{- 1}]}_{\infty}^{2 R}$
$= - G M m {[\frac{1}{x}]}_{\infty}^{2 R} = - G M m [\frac{1}{2 R} - \frac{1}{\infty}]$
$= - \frac{G M m}{2 R} (∵ \frac{1}{\infty} = 0)$

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Find ∫x=2Rx=∞GMmx2dx (G,M and m are constants)

The correct option is B −GMm2R ∫x=2Rx=∞GMmx2dx=GMm∫2R∞1x2dx =GMm∫2R∞x−2dx=GMm[x−1−1]2R∞ =−GMm[1x]2R∞=−GMm[12R−1∞] =−GMm2R(∵1∞=0)

Find $\int_{x = \infty}^{x = 2 R} \frac{G M m}{x^{2}} d x$ $(G, M$ and $m$ are constants)