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

Answer: (A)

2

Answer 2

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
$\, \, \, \, \, \, \, \, \, \, \, \, \,$ $2I = \int^{ 3 \pi / 2 }_{ \pi / 4 } \Bigg ( \frac{1}{ 1 + \cos \, x } + \frac{1}{ 1 - \cos \, x } \Bigg ) dx$
$\Rightarrow$ $\, \, \, \, \, \,$ $2I = \int^{ 3 \pi / 2 }_{ \pi / 4 } \Bigg ( \frac{2}{ 1 - \cos^2 \, x } \bigg ) \, dx$
$\Rightarrow$ $\, \, \, \, \, \,$ $I = \int^{ 3 \pi / 2 }_{ \pi / 4 } cosec^2 \, x \, dx = [ - \cot \, x ] ^{3 \pi/ 4 }_{ \pi / 4 }$
$\, \, \, \, \, \, \, \, \, \, \, \, \,$ $= \bigg [ - \cot \frac{ 3 \pi}{ 4 } + \cot \frac{\pi}{ 4 } \bigg ]$
$\, \, \, \, \, \, \, \, \, \, \, \, \,$ $= - ( - 1 ) + 1 = 2$