Question

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

• A. 2
• B. -2
• C. $ \frac{1}{2} $
• D. $ -\frac{1}{2} $

Answer 1

The Correct Option is A

Let $ I = \int^{ 3 \pi / 2 }_{ \pi / 4 } \frac{ dx }{ 1 + \cos \, x }................. ( 1) $
$\Rightarrow $ $ I = \int^{ 3 \pi / 2 }_{ \pi / 4 } \frac{ dx }{ 1 + \cos ( \pi - x ) } $
$ I = \int^{ 3 \pi / 2 }_{ \pi / 4 } \frac{ dx }{ 1 - \cos x } ...............(2) $
On adding Eqs. (i) and (ii), we get
$ \, \, \, \, \, \, \, \, \, \, \, \, \, $ $ 2I = \int^{ 3 \pi / 2 }_{ \pi / 4 } \Bigg ( \frac{1}{ 1 + \cos \, x } + \frac{1}{ 1 - \cos \, x } \Bigg ) dx $
$\Rightarrow $ $ \, \, \, \, \, \, $ $ 2I = \int^{ 3 \pi / 2 }_{ \pi / 4 } \Bigg ( \frac{2}{ 1 - \cos^2 \, x } \bigg ) \, dx $
$\Rightarrow $ $ \, \, \, \, \, \, $ $ I = \int^{ 3 \pi / 2 }_{ \pi / 4 } cosec^2 \, x \, dx = [ - \cot \, x ] ^{3 \pi/ 4 }_{ \pi / 4 } $
$ \, \, \, \, \, \, \, \, \, \, \, \, \, $ $ = \bigg [ - \cot \frac{ 3 \pi}{ 4 } + \cot \frac{\pi}{ 4 } \bigg ]$
$ \, \, \, \, \, \, \, \, \, \, \, \, \, $ $ = - ( - 1 ) + 1 = 2 $