A) \[\frac{1}{4}{{\sin }^{-1}}\left( \frac{4{{x}^{3}}}{3} \right)+c\]
B) \[\frac{1}{3}{{\sin }^{-1}}\left( \frac{4{{x}^{3}}}{3} \right)+c\]
C) \[\frac{1}{4}{{\sin }^{-1}}{{x}^{3}}+c\]
D) \[\frac{1}{3}{{\sin }^{-1}}{{x}^{3}}+c\]
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{\frac{3{{x}^{2}}}{\sqrt{9-16{{x}^{6}}}}dx=\int_{{}}^{{}}{\frac{3{{x}^{2}}}{\sqrt{{{(3)}^{2}}-{{(4{{x}^{3}})}^{2}}}}\,dx}}\] Put \[4{{x}^{3}}=t\Rightarrow 12{{x}^{2}}dx=dt,\] then it reduces to \[\frac{1}{4}\int_{{}}^{{}}{\frac{dt}{\sqrt{{{(3)}^{2}}-{{t}^{2}}}}=\frac{1}{4}.\frac{1}{1}{{\sin }^{-1}}\left( \frac{t}{3} \right)+c=\frac{1}{4}{{\sin }^{-1}}\left( \frac{4{{x}^{3}}}{3} \right)}+c\].
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