A) \[{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{3/2}}+c\]
B) \[\frac{2}{3}{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{3/2}}+c\]
C) \[\frac{3}{2}{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{3/2}}+c\]
D) \[\frac{3}{2}{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{2/3}}+c\]
Correct Answer: B
Solution :
Put \[x=a{{(\sin \theta )}^{2/3}}\Rightarrow dx=\frac{2}{3}a{{(\sin \theta )}^{-1/3}}\cos \theta \,d\theta \] \ \[\int_{{}}^{{}}{\sqrt{\frac{x}{{{a}^{3}}-{{x}^{3}}}}\,dx}=\int_{{}}^{{}}{\frac{{{a}^{1/2}}{{(\sin \theta )}^{1/3}}\frac{2}{3}a\,{{(\sin \theta )}^{-1/3}}\cos \theta }{\sqrt{{{a}^{3}}-{{a}^{3}}{{\sin }^{2}}\theta }}}\,d\theta \] \[=\frac{2}{3}{{a}^{3/2}}\int_{{}}^{{}}{\frac{\cos \theta \,d\theta }{{{a}^{3/2}}\sqrt{1-{{\sin }^{2}}\theta }}}=\frac{2}{3}{{\sin }^{-1}}{{\left( \frac{x}{a} \right)}^{3/2}}+c\].
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