I=$\int\limits_2^5 {\frac{{[{x^3}]}}{{[ - {x^3} + 21{x^2} - 147x + 343] + [{x^3}]}}dx} $

Evaluate I if [  ] represents greatest integer function.

$I = \int\limits_2^5 {\frac{{[{x^3}]}}{{[{{(7 - x)}^3}] + [{x^3}]}}dx} $ .........(*)

$I = \int\limits_2^5 {\frac{{[{{(7 - x)}^3}]}}{{[{{\{ 7 - (7 - x)\} }^3}] + [{{(7 - x)}^3}]}}dx} $, Using one of the properties of definite integration.

$I = \int\limits_2^5 {\frac{{[{{(7 - x)}^3}]}}{{[{x^3}] + [{{(7 - x)}^3}]}}dx} $ .........(#)

Adding (*) & (#),

$2I = \int\limits_2^5 {dx}  = 3$

$\therefore I = 1.5$