Question

If $F\left(x\right)={\int }_0^x\frac{\cos t}{(1+t^2)}dt,0\le x\le 2\pi$. Then

• A. $F$ is increasing in $(\frac{\pi }{2},\frac{3\pi }{2})$ and decreasing in $(0,\frac{\pi }{2})$ and $(\frac{3\pi }{2},2\pi )$
• B. $F$ is increasing in $(0,\pi )$ and decreasing in $(\pi ,2\pi )$
• C. $F$ is increasing $(\pi ,2\pi )$ and decreasing in $(0,\pi )$
• D. $F$ is increasing in $(0,\frac{\pi }{2})$ and $(\frac{3\pi }{2},2\pi )$ and decreasing in $(\frac{\pi }{2},\frac{3\pi }{2})$

Answer 1

Answer: (D)

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

Given function is
$F\left(x\right)={\int }_0^x\frac{\cos t}{(1+t^2)}dt,0\le x\le 2\pi$
On differentiation w.r.t $x$, apply (Leibnitz rule)
$F^{\prime }\left(x\right)=\frac{\cos x}{1+x^2}\times 1=\frac{\cos x}{1+x^2}$, where $(1+x^2)>0$
Here $\cos x>0$ $\Rightarrow$ $x\in (0,\frac{\pi }{2})\cup (\frac{3\pi }{2},2\pi )$
and $\cos x<0$ $\Rightarrow$ $(\frac{\pi }{2},\frac{3\pi }{2})$
So, $F$ is increasing in $(0,\frac{\pi }{2})$ and $(\frac{3\pi }{2},2\pi )$ and decreasing in $(\frac{\pi }{2},\frac{3\pi }{2})$