Explanation

(c)
Put $x = a\ \left( \sin\theta \right)^{\frac{2}{3}}$
$\Rightarrow dx = \frac{2}{3}\ a\left( \sin\theta \right)^{\frac{- 1}{3}}\cos\theta\ d\theta$
$\therefore\int_{}^{}\sqrt{\frac{x}{a^{3} - x^{3}}}\ dx = \int_{}^{}\frac{a^{\frac{1}{2}}{(\sin{\theta)}}^{\frac{1}{3}}.\frac{2}{3}\ a\ {(\sin{\theta)}}^{\frac{- 1}{3}}\cos\theta}{\sqrt{a^{3} - a^{3}\sin^{2}\theta}}d\theta$
$= \frac{2}{3}\ a^{\frac{3}{2}}\int_{}^{}\frac{\cos\theta}{a^{\frac{3}{2}}\sqrt{1 - \sin^{2}\theta}}\ d\theta$
$= \frac{2}{3}\sin^{- 1}{\left( \frac{x}{a} \right)^{\frac{3}{2}} + c = g(x) + c}$
$\therefore g(x) = \frac{2}{3}\sin^{- 1}\sqrt{\frac{x^{3}}{a^{3}}}$