∫π/43π/4 dx1+cos⁡x=

I=∫π/43π/4 dx1+cos⁡x=∫π/43π/4 dx1+cos⁡3π4+π4−x =∫π/43π/4 dx1−cos⁡x2I=∫π/43π/4 11+cos⁡x+11−cos⁡xdx=∫π/43π/4 21−cos2⁡xdx=2∫π/43π/4 cosec2⁡xdx=−[2cot⁡x]π/43π/4=−2[−1−1]=4⇒I=2

∫π/43π/4 dx1+cos⁡x=

I=∫π/43π/4 dx1+cos⁡x=∫π/43π/4 dx1+cos⁡3π4+π4−x =∫π/43π/4 dx1−cos⁡x2I=∫π/43π/4 11+cos⁡x+11−cos⁡xdx=∫π/43π/4 21−cos2⁡xdx=2∫π/43π/4 cosec2⁡xdx=−[2cot⁡x]π/43π/4=−2[−1−1]=4⇒I=2

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$\int_{π / 4}^{3 π / 4} \frac{d x}{1 + \cos x} =$

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$π - 2$

$π + 2$

$\frac{π}{4}$

answer is D.

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Detailed Solution

$I = \int_{π / 4}^{3 π / 4} \frac{d x}{1 + \cos x} = \int_{π / 4}^{3 π / 4} \frac{d x}{1 + \cos (\frac{3 π}{4} + \frac{π}{4} - x)} = \int_{π / 4}^{3 π / 4} \frac{d x}{1 - \cos x}$

$\begin{array}{l} 2 I = \int_{π / 4}^{3 π / 4} (\frac{1}{1 + \cos x} + \frac{1}{1 - \cos x}) d x \\ = \int_{π / 4}^{3 π / 4} \frac{2}{1 - \cos^{2} x} d x = 2 \int_{π / 4}^{3 π / 4} {cosec}^{2} x d x \\ = - [2 \cot x]_{π / 4}^{3 π / 4} = - 2 [- 1 - 1] = 4 \Rightarrow I = 2 \end{array}$