Question

$f\frac{ \sqrt{x} }{ \sqrt{ {a}^{3 - {x}^{3} } } } dx$
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Answer 1

Step-by-step explanation:

$\\ \text{ did \: you \: mean : - }\\ \tt \int\frac{ \sqrt{x} }{ \sqrt{ {a}^{3 } - {x}^{3} } } dx$

$\\ \rule{300pt}{1pt}$

To evaluate: $\\ \int \frac{\sqrt{x}}{\sqrt{a^{3}-x^{3}}} d x$

$\\ \displaystyle \tt\[ \int \frac{\sqrt{x}}{\sqrt{\left(a^{\frac{3}{2}}\right)^{2}-\left(x^{\frac{3}{2}}\right)^{2}}} d x \ldots (1)\]$

$\text{Put \( \tt x^{\frac{3}{2}}=t \)}$

$\\ \tt \[ \Rightarrow \frac{3}{2} x ^{\dfrac{1}{2}} dx = dt \]$

Therefore (1) becomes

$\[ \begin{array}{l} \displaystyle \tt \frac{2}{3} \int \frac{d t}{\sqrt{\left(a^{\frac{3}{2}}\right)^{2}-t^{2}}} \\ \\\displaystyle \tt \Rightarrow \frac{2}{3} \sin ^{-1}\left(\frac{t}{a^{\frac{3}{2}}}\right)+C \\ \\ \displaystyle \tt\Rightarrow \frac{2}{3} \sin ^{-1}\left(\frac{x ^ \frac{3}{2}}{x ^ \frac{3}{2}}\right)+C \\ \\ \displaystyle\tt \Rightarrow \frac{2}{3} \sin ^{-1} \sqrt{\frac{x^{3}}{a^{3}}}+C \end{array} \]$