MCQ
The correct evaluation of $\int_0^\pi {\left| {\,{{\sin }^4}x\,} \right|\,dx} $ is
- A$\frac{{8\pi }}{3}$
- B$\frac{{2\pi }}{3}$
- C$\frac{{4\pi }}{3}$
- ✓$\frac{{3\pi }}{8}$
✓
Answer
Correct option: D.
$\frac{{3\pi }}{8}$
d
(d) $\int_0^\pi {{\rm{ }}\left| {{\rm{ }}{{\sin }^4}x{\rm{ }}} \right|{\rm{ }}dx = 2\int_0^{\pi /2} {{{\sin }^4}x\,dx} } $
(d) $\int_0^\pi {{\rm{ }}\left| {{\rm{ }}{{\sin }^4}x{\rm{ }}} \right|{\rm{ }}dx = 2\int_0^{\pi /2} {{{\sin }^4}x\,dx} } $
Applying gamma function,
$2\int_0^{\pi /2} {{{\sin }^4}x\,dx} = 2\frac{{\Gamma (5/2).\Gamma (1/2)}}{{2.\Gamma (6/2)}} = \frac{{3\pi }}{8}$.
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