Join the Most Relevant JEE Main 2025 Test Series & get 99+ percentile! Join Now
Search any question & find its solution
Question: Answered & Verified by Expert
$\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{d x}{1+\cos x}$ is equal to
MathematicsDefinite IntegrationMHT CETMHT CET 2022 (06 Aug Shift 2)
Options:
  • A $-2$
  • B $-2-2 \sqrt{2}$
  • C $2$
  • D $-2 \sqrt{2}$
Solution:
1752 Upvotes Verified Answer
The correct answer is: $2$
$\begin{aligned} & \int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{d x}{1+\cos x}=\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{1}{2} \sec ^2 \frac{x}{2} d x=\left[\tan \frac{x}{2}\right]_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \\ & =\left(\tan \frac{3 \pi}{8}-\tan \frac{\pi}{8}\right)=\left(\cot \frac{\pi}{8}-\tan \frac{\pi}{8}\right)=\frac{\cos ^2 \frac{\pi}{8}-\sin ^2 \frac{\pi}{8}}{\sin \frac{\pi}{8} \cdot \cos \frac{\pi}{8}} \\ & =\frac{2 \cos \frac{\pi}{4}}{\sin \frac{\pi}{4}}=2\end{aligned}$

Looking for more such questions to practice?

Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.