_0^ 2 , , d ,x ^6 x + , ^6 ,x is equal to : - Vidyadip

MCQ

$\int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{d\,x}}{{{{\cos }^6}x + \,{{\sin }^6}\,x}}}$ is equal to :

A
zero
✓
$\pi$
C
$\pi /2$
D
$2 \pi $

✓

Answer

Correct option: B.

$\pi$

b
$I = \int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{d\,x}}{{1\,\, - \,\,3\,{{\sin }^2}\,x\,\,{{\cos }^2}\,x}}}$

$= \int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{d\,x}}{{1\,\, - \,\,{\textstyle{3 \over 4}}\,{{\sin }^2}\,2x}}}$

$= 2 \int\limits_0^\pi {\,\,\frac{{d\,t}}{{4\,\, - \,\,3\,{{\sin }^2}\,t}}}$

where $2x = t$

$= 4 \, \int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{d\,t}}{{4\,\, - \,\,3\,{{\sin }^2}\,t}}}$ etc.

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