A) \[\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}+\frac{3\pi }{8} \right)\, \right|+c\]
B) \[\frac{1}{\sqrt{2}}\log \left| \cot \left( \frac{x}{2} \right)\, \right|+c\]
C) \[\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}-\frac{3\pi }{8} \right)\, \right|+c\]
D) \[\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}-\frac{\pi }{8} \right)\, \right|+c\]
Correct Answer: A
Solution :
We have, \[I=\int_{{}}^{{}}{\frac{dx}{\cos x-\sin x}=\frac{1}{\sqrt{2}}\int_{{}}^{{}}{\frac{{{d}^{2}}}{\cos \left( \frac{\pi }{4}+x \right)}}}\] \[I=\frac{1}{\sqrt{2}}\int_{{}}^{{}}{\sec \left( x+\frac{\pi }{4} \right)}\,dx=\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{\pi }{4}+\frac{x}{2}+\frac{\pi }{8} \right)\, \right|+c\] \[I=\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}+\frac{3\pi }{8} \right)\, \right|+c\].
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