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Question: Answered & Verified by Expert
$\int \frac{1}{\cos x(1+\cos x)} d x=$
MathematicsIndefinite IntegrationJEE Main
Options:
  • A $\log (\sec x+\tan x)+2 \tan \frac{x}{2}+c$
  • B $\log (\sec x+\tan x)-2 \tan \frac{x}{2}+c$
  • C $\log (\sec x+\tan x)+\tan \frac{x}{2}+c$
  • D $\log (\sec x+\tan x)-\tan \frac{x}{2}+c$
Solution:
2290 Upvotes Verified Answer
The correct answer is: $\log (\sec x+\tan x)-\tan \frac{x}{2}+c$
$\begin{aligned} \int \frac{1}{\cos x(1+\cos x)} d x & =\int \frac{d x}{\cos x}-\int \frac{d x}{1+\cos x} \\ & =\int \sec x d x-\frac{1}{2} \int \sec ^2 \frac{x}{2} d x \\ & =\log (\sec x+\tan x)-\tan \frac{x}{2}+c .\end{aligned}$

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