What is the integration of \[ \int {\frac{{\sqrt x }}{{\sqrt {{a^3} – {x^3}} }}} dx\] ~~~~~[NCERT Exemp. Q. 24,Page 165 ]

Let \[ I = \int {\frac{{\sqrt x }}{{\sqrt {{a^3} – {x^3}} }}} dx = \int {\frac{{\sqrt x }}{{\sqrt {{{\left( {{a^{3/2}}} \right)}^2} – {{\left( {{x^{3/2}}} \right)}^2}} }}} \]


Let’s put \[ {x^{3/2}} = t \Rightarrow \frac{3}{2}{x^{1/2}}dx = dt\]

therefore,\[ I = \frac{2}{3}\int {\frac{{dt}}{{\sqrt {{{\left( {{a^{3/2}}} \right)}^2} – {t^2}} }}} = \frac{2}{3}{\sin ^{ – 1}}\frac{t}{{{a^{3/2}}}} + C\]


\[ = \frac{2}{3}{\sin ^{ – 1}}\frac{{{x^{3/2}}}}{{{a^{3/2}}}} + C = \frac{2}{3}{\sin ^{ – 1}}\sqrt {\frac{{{x^3}}}{{{a^3}}}} + C\]

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