MCQ
$\int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{d\,x}}{{{{\cos }^6}x + \,{{\sin }^6}\,x}}}$ is equal to :
- Azero
- ✓$\pi$
- C$\pi /2$
- D$2 \pi $
✓
Answer
Correct option: B.
$\pi$
b
$I = \int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{d\,x}}{{1\,\, - \,\,3\,{{\sin }^2}\,x\,\,{{\cos }^2}\,x}}}$
$I = \int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{d\,x}}{{1\,\, - \,\,3\,{{\sin }^2}\,x\,\,{{\cos }^2}\,x}}}$
$= \int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{d\,x}}{{1\,\, - \,\,{\textstyle{3 \over 4}}\,{{\sin }^2}\,2x}}}$
$= 2 \int\limits_0^\pi {\,\,\frac{{d\,t}}{{4\,\, - \,\,3\,{{\sin }^2}\,t}}}$
where $2x = t$
$= 4 \, \int\limits_0^{\frac{\pi }{2}} {\,\,\frac{{d\,t}}{{4\,\, - \,\,3\,{{\sin }^2}\,t}}}$ etc.
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