Explanation

(d)
Given, $I = \left| \int_{2}^{5}\frac{\sin{xdx}}{(1 + x^{2})} \right|$
$\leq \int_{2}^{5}{\left| \frac{\sin x}{1 + x^{2}} \right|dx}$
$\leq \int_{2}^{5}{\frac{1}{1 + x^{2}}dx \leq \int_{2}^{5}{\frac{1}{x^{2}}dx}}$
$= \left\lbrack \frac{x^{- 1}}{- 1} \right\rbrack_{2}^{5} = - \left\lbrack \frac{1}{5} - \frac{1}{2} \right\rbrack$
$\Rightarrow I \leq \frac{3}{10}$