A) 0
B) \[\frac{\sqrt{e\pi }}{6}\]
C) \[\frac{\sqrt{e\pi }}{6}-\frac{4}{3\sqrt{3}}\]
D) \[\sqrt{e}\left( \frac{\pi }{6}-\frac{2}{\sqrt{3}} \right)+1\]
Correct Answer: D
Solution :
Given integration can be written as \[\int\limits_{0}^{1/2}{{{e}^{x}}}\left[ \left( {{\sin }^{-1}}x+\frac{1}{\sqrt{1-{{x}^{2}}}} \right)-\left( \frac{1}{\sqrt{1-{{x}^{2}}}}+\frac{x}{\sqrt{{{(1-{{x}^{2}})}^{3/2}}}} \right) \right]dx\]\[=\left[ {{e}^{x}}\left( {{\sin }^{-1}}x+\frac{x}{\sqrt{1-{{x}^{2}}}} \right) \right]_{0}^{1/2}\] \[(By\int_{{}}^{{}}{{{e}^{x}}}\left( f(x)+f'(x) \right)dx={{e}^{x}}f(x))\] \[=\sqrt{e}\left( \frac{\pi }{6}-\frac{2}{\sqrt{3}} \right)+1\]
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