A) \[2{{\sin }^{-1}}(\sqrt{2}\sin x)+c\]
B) \[-2{{\sin }^{-1}}\left( \sqrt{2}\sin \frac{x}{2} \right)+c\]
C) \[2{{\sin }^{-1}}\left( \sqrt{2}\sin \frac{x}{2} \right)+c\]
D) \[\frac{1}{2}{{\sin }^{-1}}(\sqrt{2}\sin x)+c\]
Correct Answer: C
Solution :
Let\[I=\int{\sqrt{1+\sec x}}dx=\int{\sqrt{\frac{\cos x+1}{\cos x}}dx}\] \[=\int{\sqrt{\frac{2{{\cos }^{2}}\frac{x}{2}}{1-2{{\sin }^{2}}\frac{x}{2}}}dx}\] \[=\int{\frac{\sqrt{2}\cos \frac{x}{2}}{\sqrt{1-{{\left( \sqrt{2}\sin \frac{x}{2} \right)}^{2}}}}dx}\] Let \[\sqrt{2}\sin \frac{x}{2}=t\] \[\Rightarrow \] \[\sqrt{2}\cos \frac{x}{2}.\frac{1}{2}dx=dt\] \[\Rightarrow \] \[\sqrt{2}\cos \frac{x}{2}dx=2dt\] \[\therefore \] \[I=\int{\frac{2dt}{\sqrt{1-{{t}^{2}}}}=2{{\sin }^{-1}}(t)+c}\] \[=2{{\sin }^{-1}}\left( \sqrt{2}\sin \frac{x}{2} \right)+c\]
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