A) \[x+\cos x+c\]
B) \[2{{\cos }^{2}}\frac{x}{2}+c\]
C) \[\frac{1}{3}{{\left( \cos \frac{x}{2}-\frac{x}{2} \right)}^{3}}+c\]
D) \[x-\cos x+c\]
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{{{\left( \cos \frac{x}{2}-\sin \frac{x}{2} \right)}^{2}}dx}\] \[=\int_{{}}^{{}}{\left\{ {{\cos }^{2}}\frac{x}{2}+{{\sin }^{2}}\frac{x}{2}-2\sin \frac{x}{2}\cos \frac{x}{2} \right\}}dx\] \[=\int_{{}}^{{}}{(1-\sin x)\,dx}=x+\cos x+c.\]
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