A) \[{{\cosh }^{-1}}(\sin x+\cos x)+c\]
B) \[{{\sinh }^{-1}}(\sin x+\cos x)+c\]
C) \[-{{\cosh }^{-1}}(\sin x+\cos x)+c\]
D) \[-{{\sinh }^{-1}}(\sin x+\cos x)+c\]
Correct Answer: A
Solution :
\[I=\int_{{}}^{{}}{\frac{\cos x-\sin x}{\sqrt{\sin 2x}}}\,dx\]\[=\int_{{}}^{{}}{\frac{\cos x-\sin x}{\sqrt{{{(\sin x+\cos x)}^{2}}-1}}}\,dx\] Put \[\sin x+\cos x=t\Rightarrow (\cos x-\sin x)\,dx=dt\] \[I=\int_{{}}^{{}}{\frac{dt}{\sqrt{{{t}^{2}}-1}}}={{\cosh }^{-1}}t+c={{\cosh }^{-1}}(\sin x+\cos x)+c\].
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