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Question: Answered & Verified by Expert
$\int \frac{\cos x-1}{\cos x+1} d x=$
MathematicsIndefinite IntegrationJEE Main
Options:
  • A $2 \tan \frac{x}{2}-x+c$
  • B $\frac{1}{2} \tan \frac{x}{2}-x+c$
  • C $x-\frac{1}{2} \tan \frac{x}{2}+c$
  • D $x-2 \tan \frac{x}{2}+c$
Solution:
2231 Upvotes Verified Answer
The correct answer is: $x-2 \tan \frac{x}{2}+c$
$\begin{aligned} & \int \frac{\cos x-1}{\cos x+1} d x=-\int \tan ^2 \frac{x}{2} d x \\ & =-\int\left(\sec ^2 \frac{x}{2}-1\right) d x=\int\left(1-\sec ^2 \frac{x}{2}\right) d x=x-2 \tan \frac{x}{2}+c\end{aligned}$

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