Explanation

(a)
Let $I = \int_{}^{}\frac{a^{\frac{x}{2}}}{\sqrt{a^{- x} - a^{x}}}dx = \int_{}^{}\frac{a^{x}}{\sqrt{1 - a^{2x}}}dx$
Put $a^{x} = t\ \ \ \Rightarrow \ \ \ a^{x}\log{a\ dx = dt}$
$\therefore\ I = \frac{1}{\log a}\int_{}^{}\frac{1}{\sqrt{1 - t^{2}}}dt = \frac{1}{\log a}\sin^{- 1}{t + c}$
$= \frac{1}{\log a}\sin^{- 1}{\left( a^{x} \right) + c}$