A) \[\sqrt{2}\log \left( \sec \frac{x}{2}+\tan \frac{x}{2} \right)+K\]
B) \[\frac{1}{\sqrt{2}}\log \left( \sec \frac{x}{2}+\tan \frac{x}{2} \right)+K\]
C) \[\log \left( \sec \frac{x}{2}+\tan \frac{x}{2} \right)+K\]
D) None of these
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{\frac{1}{\sqrt{1+\cos x}}}\,dx=\int_{{}}^{{}}{\frac{dx}{\sqrt{2{{\cos }^{2}}(x/2)}}}=\frac{1}{\sqrt{2}}\int_{{}}^{{}}{\sec \frac{x}{2}\,dx}\] \[=\frac{1}{\sqrt{2}}\left\{ \log \left( \sec \frac{x}{2}+\tan \frac{x}{2} \right) \right\}.\frac{1}{1/2}=\sqrt{2}\log \left( \sec \frac{x}{2}+\tan \frac{x}{2} \right)+K\].
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