A) 0
B) \[\pi /2\]
C) \[\frac{\pi (a+b)}{a-b}\]
D) \[\frac{\pi }{2}({{a}^{2}}-{{b}^{2}})\]
Correct Answer: A
Solution :
[a] \[\int\limits_{0}^{2\pi }{\log \left( \frac{a+b\sec x}{a-b\sec x} \right)dx}\] \[=2\int\limits_{0}^{\pi }{\log \left( \frac{a+b\sec x}{a-b\sec x} \right)dx}\] \[=2\int\limits_{0}^{\pi }{\log (a+b\,\,\sec \,\,x)dx-2\int\limits_{0}^{\pi }{\log (a-b\sec (\pi -x))dx}}\] \[=2\int\limits_{0}^{\pi }{\log (a+b\,\,\sec \,\,x)dx-2\int\limits_{0}^{\pi }{\log (a+b\sec \,x)dx=0}}\]
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