Question

# Calculate $$\int{\dfrac{\sqrt x}{\sqrt{a^3 - x^3}}} dx$$.

Solution

## Consider the given integral. $$I=\int{\dfrac{\sqrt{x}}{\sqrt{{{a}^{3}}-{{x}^{3}}}}dx}$$   Put $$t=\dfrac{{{x}^{\frac{3}{2}}}}{{{a}^{\frac{3}{2}}}}$$ $$dt=\dfrac{1}{{{a}^{\frac{3}{2}}}}\times \dfrac{3}{2}\sqrt{x}dx$$ $$\sqrt{x}dx=\dfrac{2}{3}{{a}^{\frac{3}{2}}}dt$$ $$I=\dfrac{2}{3}\int{\dfrac{1}{\sqrt{1-{{t}^{2}}}}dt}$$ $$I=\dfrac{2}{3}{{\sin }^{-1}}t+C$$   On putting the value of $$t$$ in above expression, we get $$I=\dfrac{2}{3}{{\sin }^{-1}}{{\left( \dfrac{x}{a} \right)}^{\frac{3}{2}}}+C$$   Hence, this is the required value of the given integral.Mathematics

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