A) \[\frac{1}{2}\sin x\sqrt{4-{{\sin }^{2}}x}-2{{\sin }^{-1}}\left( \frac{1}{2}\sin x \right)+c\]
B) \[\frac{1}{2}\sin x\sqrt{4-{{\sin }^{2}}x}+2{{\sin }^{-1}}\left( \frac{1}{2}\sin x \right)+c\]
C) \[\frac{1}{2}\sin x\sqrt{4-{{\sin }^{2}}x}+{{\sin }^{-1}}\left( \frac{1}{2}\sin x \right)+c\]
D) None of these
Correct Answer: B
Solution :
Putting \[\sin x=t\Rightarrow \cos x\,dx=dt,\] we get \[\int_{{}}^{{}}{\cos x\sqrt{4-{{\sin }^{2}}x\,}dx}=\int_{{}}^{{}}{\sqrt{4-{{t}^{2}}}dt=\int_{{}}^{{}}{\sqrt{{{(2)}^{2}}-{{t}^{2}}}dt}}\] \[=\frac{t}{2}\sqrt{4-{{t}^{2}}}+\frac{4}{2}{{\sin }^{-1}}\frac{t}{2}+c\] \[=\frac{1}{2}\sin x\sqrt{4-{{\sin }^{2}}x}+2{{\sin }^{-1}}\left( \frac{1}{2}\sin x \right)+c.\]
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