Question

$\int \frac{\sec xcosecx}{2\cot x-\sec x\cos ecx}dx$ 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)

$\int \frac{\sec x\cos ecx}{2\cot x-\sec x\cos ecx}dx$
$=\int \frac{\frac{1}{\cos x\sin x}}{\frac{2\cos x}{\sin x}-\frac{1}{\sin x\cos x}}dx$
$=\int \frac{dx}{2{\cos }^{2}x-1}$
$=\int \frac{dx}{{\cos }^{2}x-{\sin }^{2}x}=\int \sec 2xdx$
$=\frac{1}{2}\log |\sec 2x+\tan 2x|+c$