Explanation

(a)
Let $I = \int_{}^{}{\frac{3^{x}}{\sqrt{9^{x} - 1}}\ dx} = \frac{3^{x}dx}{\sqrt{\left( 3^{x} \right)^{2} - 1}}$
Put $3^{x} = t\ \Rightarrow 3^{x}\log{3dx = dt}$
$\Rightarrow 3^{x}dx = \frac{dt}{\log 3}$
$\therefore I = \frac{1}{\log 3}\int_{}^{}\frac{dt}{\sqrt{t^{2} - 1\ }}$
$= \frac{1}{\log 3}\log{\left\lbrack \ t + \sqrt{t^{2} - 1\ }\ \right\rbrack +}c$
$= \frac{1}{\log 3}\log{\left\lbrack \ 3^{x} + \sqrt{9^{x} - 1}\ \right\rbrack + c}$