Question

The value of $ \int^{ \pi / 2}_0 \frac{ dx }{ 1 + \tan^3 \, x } \, is \, $

• A. 0
• B. 1
• C. $\frac{ \pi }{ 2 } $
• D. $\frac{ \pi }{ 4 } $

Answer 1

The Correct Option is D

Let $ I = \int^{ \pi / 2 }_0 \frac{ 1}{ 1 + \tan ^3 x } \, dx = \int^{ \pi/ 2}_0 \frac {1}{ 1 + \frac{ \sin^3 \, x }{ \cos ^3 \, x } } \, dx $
$\Rightarrow $ $ I = \int^{ \pi / 2 }_0 \frac { \cos^3 \, x }{ \cos^3 \, x + \sin^3 \, x}$
$\Rightarrow $ $ I = \int^{ \pi / 2 }_0 \frac{ \cos^3 \bigg ( \frac{\pi}{2 } - x \bigg ) }{ \cos^3 \bigg ( \frac{\pi}{2 } - x \bigg ) + \sin ^3 \bigg ( \frac { \pi }{2 } - x \bigg ) } \, dx $
$\Rightarrow $ $ I = \int^{ \pi / 2 } _ 0 \frac{ \sin^3 \, x }{ \sin^3 \, x + \cos^3 \, x} \, dx ..............(2) $
On adding Eqs. (i) and (ii), we get
$ 2I = \int^{ \pi / 2 }_ 0 \, 1 \, dx $ $\Rightarrow $ $ 2 \, I = [ x ]^{ \pi / 2 }_ 0 = \pi/ 2 $ $\Rightarrow $ $ I = \pi / 4 $