Evaluate `int_0^sqrt3 (1/(1+x^2)) sin^(-1)((2x)/(1+x^2)) dx`

Explanation

For`0ltxlt1, "sin"^(-1)(2x)/(1+x^(2))=2tan^(-1)x`
For `xgt1, "sin"^(-1)(2x)/(1+x^(2))="sin"^(-1)(2/x)/(1+1/(x^(2)))=2tan^(-1)(1//x)=2cot^(-1)x`
Hence the given integral is
`int_(0)^(sqrt(3))("sin"^(-1)((2x)/(1+x^(2))))/((1+x^(2))) dx=int_(0)^(1)2/(1+x^(2))(tan^(-1)x)dx+int_(1)^(sqrt(3))(2cot^(-1)x)/(1+x^(2))dx`
`=[(tan^(-1)x)^(2)]_(0)^(1)-[(cot^(-)x)^(2)]_(1)^(sqrt(3))=(pi^(2))/16-((pi^(2))/36-(pi^(2))/16)`
`=(pi^(2))/8-(pi^(2))/36=(7pi^(2))/72`