Question

If $f(x)=\left|\begin{array}{ccc}\cos x& 1& 0\\ 1& 2\cos x& 1\\ 0& 1& 2\cos x\end{array}\right|$, then $\underset{0}{\overset{\pi /2}{\int }}f(x)dx=$

• A. $\frac{1}{4}$
• B. $-\frac{1}{3}$
• C. $\frac{1}{2}$
• D. 1

Answer 1

Answer: (B)

$f(x)=\left|\begin{array}{ccc}\cos x& 1& 0\\ 1& 2\cos x& 1\\ 0& 1& 2\cos x\end{array}\right|$
$=\cos x\left(4{\cos }^{2}x-1\right)-1\left(2{\cos }^{2}x\right)+0$
$=4{\cos }^{3}x-\cos x-2\cos x=4{\cos }^{3}x-3\cos x=\cos 3x$
$\therefore \underset{0}{\overset{\frac{\pi }{2}}{\int }}f(x)=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\cos 3x={\left(\frac{\sin 3x}{3}\right)}_0^{\frac{\pi }{2}}=-\frac{1}{3}$