$\int \large\frac{\sqrt x }{\sqrt {a^3-x^3}}$$dx$
$x^{3/2} x=t$
differentiate with x
$\large\frac{3}{2} $$x^{1/2} dx=dt$
=> $\int \large\frac{2}{3} \large\frac{dt}{\sqrt {(a^{3/2})^2 -(t)^2}}$
$\large\frac{2}{3} $$ \sin ^{-1} \bigg( \large\frac{t}{a^{3/2}}\bigg ) $$+c$
Hence a is the correct answer.