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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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