Explanation

(a)
$\because x^{2} < x^{\frac{\pi}{2}} < x$
$\Rightarrow 1 + x^{2} < 1 + x^{\frac{\pi}{2}} < 1 + x$
$\Rightarrow \frac{1}{1 + x} < \frac{1}{1 + x^{\frac{\pi}{2}}} < \frac{1}{1 + x^{2}}$
$\Rightarrow \int_{0}^{1}{\frac{dx}{1 + x} < \int_{0}^{1}{\frac{1}{1 + x^{\frac{\pi}{2}}}dx < \int_{0}^{1}\frac{dx}{1 + x^{2}}}}$
$\Rightarrow \left\{ \left. \ \log(1 + x) \right\} \right.\ _{0}^{1} < I < (\tan^{- 1}{x\left. \ \ \right)_{0}^{1}}$
$\Rightarrow \log{2 < I < \frac{\pi}{4}}$