If $f: D \in R$ be such that $f(x)=\sqrt{\sin (\cos x)}+\ln \left(-2 \cos ^2 x+3 \cos x-1\right)$, then $\int\limits_{x_1}^{x_2}\left[\cos x-\frac{1}{2}\right] d x$ is equal to, where $x_1, x_2 \in D$ and [.] denotes the greatest integer function,
Answer & explanation
Correct answer: option 1
Clearly, $\sqrt{\sin (\cos x)}$ is defined for all $x \in R$.
$\ln \left(-2 \cos ^2 x+3 \cos x-1\right)$ is defined, if
$-2 \cos ^2 x+3 \cos x-1>0$
$\Rightarrow 2 \cos ^2 x-3 \cos x+1<0$
$\Rightarrow (2 \cos x-1)(\cos x-1)<0$
$\Rightarrow \frac{1}{2}<\cos x<1$
$\Rightarrow 0<\cos x-\frac{1}{2}<\frac{1}{2} \Rightarrow\left[\cos x-\frac{1}{2}\right]=0$
∴ $\int\limits_{x_1}^{x_2}\left[\cos x-\frac{1}{2}\right] d x=\int\limits_{x_1}^{x_2} 0 d x=0$