Find the value of `int_(-2)^(2)(sin^(-1)(sinx)+cos^(-1)(cosx))/((1+x^(2))(1+[(x^(2))/5]))dx`, where [.] represents the greatest integer function.

Explanation

`I=int_(-2)(2)(sin^(-1)(sinx)+cos^(-1)(cosx))/((1+x^(2))(1+[(x^(2))/5]))`
`sin^(-1)(sinx)` is an odd function.
Also for `-2lt xlt 2, 0le x^(2)lt4`.
`:. [(x^(2))/5]=0`
`:.I=int_(-1)^(2)(cos^(-1)(cosx))/(1+x^(2))dx`
`=int_(0)^(2)(2cos^(-1)(cosx))/(1+x^(2))dx`
`=int_(0)^(2)(2x)/(1+x^(2)dx`
`=[log_(e)(1+x^(2))]_(0)^(2)`
`=log_(e)4`