Explanation

(a)
$\int_{}^{}\frac{dx}{a^{2}\sin^{2}{x + b^{2}\cos^{2}x}}$
$= \int_{}^{}\frac{\sec^{2}{x\ dx}}{a^{2}\tan^{2}{x + b^{2}}}$
$= \frac{1}{a^{2}}\int_{}^{}\frac{dt}{t^{2} + \left( \frac{b}{a} \right)^{2}}$
Where $t = \tan x$
$= \frac{1}{a^{2}} \bullet \frac{a}{b}\tan^{- 1}{\frac{t}{b/a} + c}$
$= \frac{1}{ab}\tan^{- 1}{\frac{a(\tan{x)}}{b} + c}$