Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
The integral $\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{d x}{1+\cos x}$ is equal to
A
-1
B
-2
✓
2
D
4
✓
Answer
Correct option: C.
2
(C) Let $I =\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{d x}{1+\cos x}$ ...(i) $\therefore \quad I=\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{d x}{1+\cos (\pi-x)}$ $\ldots\left[\because \int_{ a }^{ b } f (x) d x=\int_{ a }^{ b } f ( a + b -x) d x\right]$ $\therefore \quad I=\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{d x}{1-\cos x}$ ...(ii) Adding (i) and (ii), we get $2 I =\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \frac{2}{1-\cos ^2 x} d x$ $\therefore \quad I =\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \operatorname{cosec}^2 x d x$ $=-[\cot x]_{\pi / 4}^{3 \pi / 4}=2$
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