Question

$\underset{-1}{\stackrel{2}{\int }}\left[\frac{\left[X\right]}{1+X^2}\right]dx$, where [.] denotes greatest integer function.

Answer 1

${\int }_{-1}^2\left[\frac{\left[X\right]}{X^2+1}\right]dx$

substitute $u=x^2+1\to du=2xdx$

$\frac{1}{2}\int \frac{1}{u}du$

$=\frac{\ln (x^2+1)}{2}+C$

${\int }_{-1}^2\left[\frac{\left[X\right]}{X^2+1}\right]dx=\frac{\ln 5-\ln 2}{2}$

$=0.458$