Question

Let $f(x)={\cos }^{-1}\left(\frac{2x}{1+x^2}\right),g(x)={\cot }^{-1}\left(\frac{2x}{x^2-1}\right)$ 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

Answer: 4

$f(x)=\frac{\pi }{2}-{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)=\frac{\pi }{2}-2{\tan }^{-1}x$
$g(x)=\frac{\pi }{2}-{\tan }^{-1}\left(\frac{2x}{x^2-1}\right)=\frac{\pi }{2}+2{\tan }^{-1}x$
$y=f(x)+g(x)=\pi$
$y=\pi x^2$
$\text{ Required area }=2\pi -2\underset{0}{\overset{1}{\int }}\pi x^{2}dx$
$A=2\pi -2\pi \cdot \frac{1}{3}=\frac{4\pi }{3}\simeq 4.19$
$\therefore [A]=4$