_ / 4 ^ 3 / 4 d x 1+ x is equal to - Vidyadip

MCQ

$\int_{\pi / 4}^{3 \pi / 4} \frac{d x}{1+\cos x}$ is equal to

✓
2
B
-2
C
$\frac{1}{2}$
D
$-\frac{1}{2}$

✓

Answer

Correct option: A.

2

(A)
$\int_{\pi / 4}^{3 \pi / 4} \frac{d x}{1+\cos x}$
$=\int_{\pi / 4}^{3 \pi / 4} \frac{1-\cos x}{1-\cos ^2 x} d x=\int_{\pi / 4}^{3 \pi / 4} \frac{1-\cos x}{\sin ^2 x} d x$
$=\int_{\pi / 4}^{3 \pi / 4}\left(\operatorname{cosec}^2 x-\cot x \operatorname{cosec} x\right) d x$
$=[-\cot x+\operatorname{cosec} x]_{\frac{\pi}{4}}^{\frac{3 \pi}{4}}=2$

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