MCQ
$\int_{\, - \pi /2}^{\,\pi /2} {{{\sin }^4}x{{\cos }^6}x\,dx = } $
- A$\frac{{3\pi }}{{64}}$
- B$\frac{{3\pi }}{{572}}$
- ✓$\frac{{3\pi }}{{256}}$
- D$\frac{{3\pi }}{{128}}$
$ = 2\int_0^{^{\pi /2}} {{{\sin }^4}x\,{{\cos }^6}x.\,dx} $
$\begin{matrix}
\because \int_{-a}^{a}{f(x)\,dx=2\int_{0}^{a}{f(x)\,dx,}} & \text{if }f(-x)=f(x) \\
\,\,\,\,\,=0, & \text{if }f(-x)=-f(x) \\
\end{matrix}$
Applying Gamma function, we get
$I = \frac{{2\,\Gamma 5/2\,.\,\Gamma 7/2}}{{2\,.\Gamma 6}}$
$ = \frac{{3/2.1/2.\sqrt {\pi .} 5/2.3/2.1/2.\sqrt \pi }}{{5.4.3.2.1}}$
$ = \frac{{3\pi }}{{{2^8}}} = \frac{{3\pi }}{{256}}$.
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$f (\theta)=\left|\begin{array}{ccc}-\sin ^{2} \theta & -1-\sin ^{2} \theta & 1 \\ -\cos ^{2} \theta & -1-\cos ^{2} \theta & 1 \\ 12 & 10 & -2\end{array}\right|$ are $m$ and $M$ respectively, then the ordered pair $( m , M )$ is equal to