Question

$ \int{\frac{\sec x cosec x}{2\cot x-\sec x\cos ecx}}dx $ is equal to

• 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 $

Answer 1

Answer: (C)

$ \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 $