Question

If $f\left(x\right)=\left|\begin{array}{ccc}\sin x+\sin 2x+\sin 3x& \sin 2x& \sin 3x\\ 3+4\sin x& 3& 4\sin x\\ 1+\sin x& \sin x& 1\end{array}\right|$, then the value of ${\int }_0^{\frac{\pi }{2}}f\left(x\right)dx$, is

• A. $3$
• B. $\frac{2}{3}$
• C. $\frac{1}{3}$
• D. $0$

Answer 1

Answer: (C)

$\frac{1}{3}$

Using Determinant properties:
$C_1\to C_1-(C_2+C_3)$

We get,
$\left|\begin{array}{ccc}\sin x& \sin 2x& \sin 3x\\ 0& 3& 4\sin x\\ 0& \sin x& 1\end{array}\right|$
$=\sin x(3-4{\sin }^{2}x)$
$=3\sin x-4{\sin }^{3}x$
$=\sin 3x$

${\int }_0^{\frac{\pi }{2}}\sin 3xdx={\frac{-\cos 3x}{3}|}_0^{\frac{\pi }{2}}=0+\frac{1}{3}$
$=\frac{1}{3}$