Question

Let $f(x)=\cos ^{-1}\left(\frac{2 x}{1+x^2}\right), g(x)=\cot ^{-1}\left(\frac{2 x}{x^2-1}\right)$ where $x \in(-1,1)$. If area bounded by the curves $y=f(x)+g(x)$ and $y=\pi x^2$ is A then find the value of $[A]$.
[Note: $[ K ]$ denotes greatest integer less than or equal to $K$.]

Answer 1

$f ( x )=\frac{\pi}{2}-\sin ^{-1}\left(\frac{2 x }{1+ x ^2}\right)=\frac{\pi}{2}-2 \tan ^{-1} x$
$g(x)=\frac{\pi}{2}-\tan ^{-1}\left(\frac{2 x}{x^2-1}\right)=\frac{\pi}{2}+2 \tan ^{-1} x$
$y=f(x)+g(x)=\pi $
$y=\pi x^2$
$\text { Required area }=2 \pi-2 \int\limits_0^1 \pi x^2 d x$
$A=2 \pi-2 \pi \cdot \frac{1}{3}=\frac{4 \pi}{3} \simeq 4.19 $
$\therefore [A]=4$