Explanation

(c)
Given, $f(x) = \left| \begin{matrix}
\sin{x + \sin{2x + \sin{3x}}} & \sin{2x} & \sin{3x} \\
3 + 4\sin x & 3 & 4\sin x \\
1 + \sin x & \sin x & 1
\end{matrix} \right|$
Applying $C_{1} \rightarrow C_{1} - \left( C_{2} + C_{3} \right)$, we get
$= \left| \begin{matrix}
\sin x & \sin{2x} & \sin{3x} \\
0 & 3 & 4\sin x \\
0 & \sin x & 1
\end{matrix} \right|$
$= \sin{x(3 - 4\sin^{2}{x) = 3\sin{x - 4\sin^{3}x}}}$
$f(x) = \sin{3x}$
$\therefore\ \int_{0}^{\pi/2}{f(x)dx = \int_{0}^{\pi/2}{\sin{3x\ dx}}}$
$= \left\lbrack - \frac{\cos{3x}}{3} \right\rbrack_{0}^{\pi/2} = \left( - \frac{1}{3} \right)\lbrack{\cos{3x\rbrack}}_{0}^{\pi/2}$
$= \left( - \frac{1}{3} \right)\left\lbrack \cos{\left( 3 \times \frac{\pi}{2} \right) - \cos 0} \right\rbrack\$
$= \left( - \frac{1}{3} \right)(0 - 1) = \frac{- 1}{3}( - 1) = \frac{1}{3}$