A) \[\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}-\frac{\pi }{8} \right) \right|+c\]
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) \[\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}+\frac{3\pi }{8} \right) \right|+c\]
Correct Answer: D
Solution :
\[I=\int{\frac{dx}{\cos x-\sin x}}\] \[=\frac{1}{\sqrt{2}}\int{\frac{dx}{\left( \frac{1}{\sqrt{2}}\cos x-\frac{1}{\sqrt{2}}\sin x \right)}}\] \[=\frac{1}{\sqrt{2}}\int{\frac{dx}{\cos \left( x+\frac{\pi }{4} \right)}}=\frac{1}{\sqrt{2}}\int{\sec \left( x+\frac{\pi }{4} \right)}\,dx\] \[=\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{\pi }{4}+\frac{x}{2}+\frac{\pi }{8} \right) \right|+c\] \[=\frac{1}{\sqrt{2}}\log \left| \tan \left( \frac{x}{2}+\frac{3\pi }{8} \right) \right|+c\]
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