A) \[\log |\sec x+\tan x|+c\]
B) \[\log |\sec x+\cos ecx|+c\]
C) \[\frac{1}{2}\log |\sec 2x+\tan 2x|+c\]
D) \[\log |\sec 2x+\cos ec2x|+c\]
E) \[\log |\sec 2x\,\cos ec2x|+c\]
Correct Answer: C
Solution :
\[\int{\frac{\sec x\cos ecx}{2\cot x-\sec x\cos ecx}}dx\] \[=\int{\frac{\frac{1}{\cos x\sin x}}{\frac{2\cos x}{\sin x}-\frac{1}{\sin x\cos x}}}dx\] \[=\int{\frac{dx}{2{{\cos }^{2}}x-1}}\] \[=\int{\frac{dx}{{{\cos }^{2}}x-{{\sin }^{2}}x}=\int{\sec 2x\,dx}}\] \[=\frac{1}{2}\log |\sec 2x+\tan 2x|+c\]
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