SATHEE: Indefinite Integration 1 Question 30

Indefinite Integration 1 Question 30

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

$(1999, 2 M)$

(a) 2

(b) -2

(d) $- \frac{1}{2}$

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Solution:

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

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

On adding Eqs. (i) and (ii), we get

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

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Indefinite Integration 1 Question 30

31. ∫π/43π/4dx1+cos⁡x is equal to

Solution:

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