If $\smallint \,\,\sqrt {\frac{x}{{{a^3} - {x^3}}}} \,\,dx = {\rm{g}}\left( x \right) + c$, then g(x) is equal to

$\frac{2}{3}\,\,{\cos ^{ - 1}}\,\,x$

$\frac{2}{3}\,\,{\sin ^{ - 1}}\,\,\left( {\frac{{{x^3}}}{{{a^3}}}} \right)$

$\frac{2}{3}\,\,{\cos ^{ - 1}}\,\,\left( {\frac{x}{a}} \right)$

JEE MATHEMATICS MCQ: INDEFINITE INTEGRATION

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Application +4 | -1

\int \,\,\sqrt {\frac{x}{{{a^3} - {x^3}}}} \,\,dx = \mathrm{g}\left( x \right) + c

, then g(x) is equal to

View answer and explanation

Correct option: C

\frac{2}{3}\,\,{\sin ^{ - 1}}\,\,\left( {\sqrt {\frac{{{x^3}}}{{{a^3}}}} } \right)

Option C is correct because the integral can be solved using trigonometric substitution, leading to an inverse sine function.