Question

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

• A. 2
• B. –2
• C. ½
• D. –1/2

Answer 1

Answer: (A)

2

Answer 2

$I = \int_{\pi/4}^{3\pi/4}{\frac{1}{1 - \cos x}dx}$ [$\because\left\lbrack \cos\left( \frac{\pi}{4} + \frac{3\pi}{4} - x \right) = - \cos x \right\rbrack$

∴ $2I = \int_{\pi/4}^{3\pi/4}{\frac{2}{1 - \cos^{2}x}dx}$

⇒ $2I = 2\int_{\pi/4}^{3\pi/4}{co\text{se}\text{c}^{2}xdx}$ ⇒ $2I = - 2\lbrack\cot x\rbrack_{\pi ⥂ / ⥂ 4}^{3\pi ⥂ / ⥂ 4} = 4$ ⇒ I = 2.