A) \[\frac{2}{3}{{\cos }^{-1}}x\]
B) \[\frac{2}{3}{{\sin }^{-1}}\left( \frac{{{x}^{3}}}{{{a}^{3}}} \right)\]
C) \[\frac{2}{3}{{\sin }^{-1}}\left( \sqrt{\frac{{{x}^{3}}}{{{a}^{3}}}} \right)\]
D) \[\frac{2}{3}{{\cos }^{-1}}\left( \frac{x}{a} \right)\]
Correct Answer: C
Solution :
Let \[I=\int_{{}}^{{}}{\sqrt{\frac{x}{{{a}^{3}}-{{x}^{3}}}}dx}\] Put \[{{x}^{3/2}}=t\Rightarrow \frac{3}{2}{{x}^{1/2}}dx=dt\] \[\therefore \] \[I=\frac{2}{3}\int_{{}}^{{}}{\frac{dt}{\sqrt{{{a}^{3}}-{{t}^{2}}}}}\] \[=\frac{2}{3}{{\sin }^{-1}}\frac{t}{{{a}^{3/2}}}+c\] \[=\frac{2}{3}{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{3/2}}+c\] \[=\frac{2}{3}{{\sin }^{-1}}\left( \sqrt{\frac{{{x}^{3}}}{{{a}^{3}}}} \right)+c\] But \[I=g(x)+c\] \[\therefore \] \[g(x)=\frac{2}{3}{{\sin }^{-1}}\left( \sqrt{\frac{{{x}^{3}}}{{{a}^{3}}}} \right)\]
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