If $F(x)=\int_{0}^{x} \frac{\cos t}{\left(1+t^{2}\right)} d t, 0 \leq x \leq 2 \pi .$ Then

• A

  $F$ is increasing in $\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$ and decreasing in $\left(0, \frac{\pi}{2}\right)$ and $\left(\frac{3 \pi}{2}, 2 \pi\right)$
• B

  $F$ is increasing in $(0, \pi)$ and decreasing in $(\pi, 2 \pi)$
• C

  $F$ is increasing $(\pi, 2 \pi)$ and decreasing in $(0, \pi)$
• Correct answer
  D

  $F$ is increasing in $\left(0, \frac{\pi}{2}\right)$ and $\left(\frac{3 \pi}{2}, 2 \pi\right)$ and decreasing in $\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$

Hence, the correct answer is option: 'D'.