A) \[\log |\cos \theta -\sin \theta +\sqrt{\sin 2\theta }|+c\]
B) \[\log |\sin \theta -\cos \theta +\sqrt{\sin 2\theta }|+c\]
C) \[{{\sin }^{-1}}(\sin \theta -\cos \theta )+c\]
D) \[{{\sin }^{-1}}(\sin \theta +\cos \theta )+c\]
E) \[{{\sin }^{-1}}(\cos \theta -\sin \theta )+c\]
Correct Answer: C
Solution :
Let\[I=\int{\frac{\sin \theta +\cos \theta }{\sqrt{1+\sin 2\theta -1}}d\theta }\] \[=\int{\frac{\sin \theta +\cos \theta }{\sqrt{1-{{(\sin \theta -\cos \theta )}^{2}}}}d\theta }\] Let \[\sin \theta -\cos \theta =t\] \[\Rightarrow \] \[(\cos \theta +\sin \theta )d\theta =dt\] \[\therefore \] \[I=\int{\frac{1}{\sqrt{1-{{t}^{2}}}}}dt={{\sin }^{-1}}(\sin \theta -\cos \theta )+c\]
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