The value of _0^ /2 dx 1 + ^3 x is - Vidyadip

MCQ

The value of $\int_0^{\pi /2} {\frac{{dx}}{{1 + {{\tan }^3}x}}} $ is

A
$0$
B
$1$
C
$\frac{\pi }{2}$
✓
$\frac{\pi }{4}$

✓

Answer

Correct option: D.

$\frac{\pi }{4}$

d
(d) $I = \int_0^{\pi /2} {\frac{{dx}}{{1 + {{\tan }^3}x}} = \int_0^{\pi /2} {\frac{{{{\cos }^3}x}}{{{{\sin }^3}x + \cos {x^3}}}} } dx$ ....$(i)$

$ = \int_0^{\pi /2} {\frac{{{{\sin }^3}x}}{{{{\cos }^3}x + {{\sin }^3}x}}dx} $ .....$(ii)$

Adding $(i)$ and $(ii),$ we get

$2I = \int_0^{\pi /2} {dx \Rightarrow I = \frac{\pi }{4}.} $

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