A) \[{{\cos }^{-1}}\frac{{{x}^{2}}}{2}\]
B) \[\frac{1}{2}{{\cos }^{-1}}\frac{{{x}^{2}}}{2}\]
C) \[{{\sin }^{-1}}\frac{{{x}^{2}}}{2}\]
D) \[\frac{1}{2}{{\sin }^{-1}}\frac{{{x}^{2}}}{2}\]
Correct Answer: D
Solution :
\[\int_{{}}^{{}}{\frac{x}{\sqrt{4-{{x}^{4}}}}}\,dx=\int_{{}}^{{}}{\frac{x}{\sqrt{{{2}^{2}}-{{({{x}^{2}})}^{2}}}}}\,dx\] Putting \[{{x}^{2}}=t\Rightarrow 2x\,dx=dt,\] we get the required integral \[=\frac{1}{2}{{\sin }^{-1}}\frac{{{x}^{2}}}{2}\].
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