A) \[\log \left| \cos \theta -\sin \theta +\sqrt{\sin 2\theta } \right|\]
B) \[\log \left| \sin \theta -\cos \theta )+\sqrt{\sin 2\theta } \right|\]
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 \[I=\int{\frac{\sin \theta +\cos \theta }{\sqrt{2\sin \theta \cos \theta }}d\theta }\] \\[I=\int{\frac{\sin \theta +\cos \theta }{\sqrt{1-(1-2\sin \theta \cos \theta )}}d\theta }\] \[=\int{\frac{(\sin \theta +\cos \theta )d\theta }{\sqrt{1-({{\sin }^{2}}\theta +{{\cos }^{2}}\theta -2\sin \theta \cos \theta )}}}\] \[=\int{\frac{\sin \theta +\cos \theta }{\sqrt{1-{{(\sin \theta -\cos \theta )}^{2}}}}d\theta }\] Let \[(\sin \theta -\cos \theta )=t\] Þ \[(\cos \theta +\sin \theta )d\theta =dt\] \ \[I=\int{\frac{dt}{\sqrt{1-{{t}^{2}}}}={{\sin }^{-1}}(t)+c}={{\sin }^{-1}}(\sin \theta -\cos \theta )+c\].
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