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

Answer: (D)

$\frac{ \pi }{ 4 }$

Answer 2

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
$2I = \int^{ \pi / 2 }_ 0 \, 1 \, dx$ $\Rightarrow$ $2 \, I = [ x ]^{ \pi / 2 }_ 0 = \pi/ 2$ $\Rightarrow$ $I = \pi / 4$