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