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Question: \(\int_{}^{}{\frac{\sec x.\cos ecx}{2\cot x - \sec x\cos ecx}dx}\) is equal to-...

∫sec⁡x.cos⁡ecx2cot⁡x−sec⁡xcos⁡ecxdx\int_{}^{}{\frac{\sec x.\cos ecx}{2\cot x - \sec x\cos ecx}dx} is equal to-

A

12\frac{1}{2}ln | sec2x + tan 2x | + C

B

ln | secx + cosec x | + C

C

ln | sec x + tan x | + C

D

12\frac{1}{2}ln | sec x + cosec x | + C

Answer

12\frac{1}{2}ln | sec2x + tan 2x | + C

Explanation

Solution

I = ∫sec⁡xcos⁡ecx2cot⁡x−sec⁡xcos⁡ecxdx\int_{}^{}{\frac{\sec x\cos ecx}{2\cot x - \sec x\cos ecx}dx} = ∫dx2cos⁡2x−1\int_{}^{}\frac{dx}{2\cos^{2}x - 1}

= ∫sec⁡2xdx\int_{}^{}{\sec 2xdx} = 12\frac{1}{2}ln | sec 2x + tan 2x | + C