Q53 / 120 · VITEEE 2011 · Mathematics
Evaluate $\int_{\pi / 4}^{3 \pi / 4} \frac{1}{1+\cos x} d x$
- Correct answerA2
- B$-2$
- C$1 / 2$
- D$-1 / 2$
Solution
$$
\begin{array}{l}
\int_{\pi / 4}^{3 \pi / 4} \frac{1}{1+\cos x} d x=\int_{\pi / 4}^{3 \pi / 4} \frac{1-\cos x}{\sin ^{2} x} d x \\
=\int_{\pi / 4}^{3 \pi / 4}\left(\operatorname{cosec}^{2} x-\cot x \operatorname{cosec} x\right) d x \\
=[-\cot x+\operatorname{cosec} x]_{\pi / 4}^{3 \pi / 4} \\
=(1+\sqrt{2})-(-1+\sqrt{2})=2
\end{array}
$$
Hence, the correct answer is option: 'A'.