Question
$\text{Find} \int \frac{\sqrt{x}}{\sqrt{\text{a}^{3}} - \text{x}^{3}}\text{dx}.$
✓
Answer
$\text{I} = \int \frac{\sqrt{x}}{\sqrt{\text{a}^{3}} - \text{x}^{3}}\text{dx}$
$\text{Put x}^{3/2} = \text{t}\Rightarrow\frac{3}{2}.\text{x}^{1/2}\text{dx = dt or}\sqrt{\text{x }} dx= \frac{3}{2}\text{dt}$
$\text{I} = \frac{2}{3}\int\frac{\text{dt}}{\sqrt{\text{(a}^{3/2})^{2} - \text{t}^{2}}}$
$= \frac{2}{3}.\sin^{-1}\bigg(\frac{\text{t}}{\text{a}^{3/2}}\bigg)+\text{C}$
$= \frac{2}{3} \sin^{-1}\bigg(\frac{\text{x}^{3/2}}{\text{a}^{3/2}}\bigg)+ \text{C}$
$\text{Put x}^{3/2} = \text{t}\Rightarrow\frac{3}{2}.\text{x}^{1/2}\text{dx = dt or}\sqrt{\text{x }} dx= \frac{3}{2}\text{dt}$
$\text{I} = \frac{2}{3}\int\frac{\text{dt}}{\sqrt{\text{(a}^{3/2})^{2} - \text{t}^{2}}}$
$= \frac{2}{3}.\sin^{-1}\bigg(\frac{\text{t}}{\text{a}^{3/2}}\bigg)+\text{C}$
$= \frac{2}{3} \sin^{-1}\bigg(\frac{\text{x}^{3/2}}{\text{a}^{3/2}}\bigg)+ \text{C}$
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