A) \[\frac{1}{5}\]
B) \[\frac{2}{5}\]
C) \[\text{O}\left( 0,0,0 \right)\]
D) \[\text{A}\left( \text{1},\text{2},\text{1} \right),\text{B}\left( \text{2},\text{1},\text{3} \right)\]
Correct Answer: D
Solution :
Let\[I=\int{\sqrt{\frac{x}{{{a}^{3}}-{{x}^{3}}}}}dx\] \[=\int{\frac{\sqrt{x}}{{{({{a}^{3/2}})}^{2}}-{{({{x}^{3/2}})}^{2}}}}dx\] \[=\frac{2}{3}\int{\frac{\frac{3}{2}\sqrt{x}}{{{({{a}^{3/2}})}^{2}}-{{({{x}^{3/2}})}^{2}}}}d({{x}^{3/2}})\] \[=\frac{2}{3}\int{\frac{1}{{{({{a}^{3/2}})}^{2}}-{{({{x}^{3/2}})}^{2}}}}d({{x}^{3/2}})\] \[=\frac{2}{3}\int{\frac{dt}{\sqrt{{{({{a}^{3/2}})}^{2}}-{{t}^{2}}}}}\,\,\,where\,\,\,{{x}^{3/2}}=t\] \[=\frac{2}{3}{{\sin }^{-1}}\left( \frac{t}{{{a}^{3/2}}} \right)+C\] \[=\frac{2}{3}{{\sin }^{-1}}\left( \frac{{{x}^{3/2}}}{{{a}^{3/2}}} \right)+C\]
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