Evaluate: `int_0^(sqrt(3))1/(1+x^2)sin^(-1)((2x)/(1+x^2))dx`

Explanation

For `0 lt x lt 1, sin ^{-1} frac{2 x}{1+x^{2}}=2 tan ^{-1} x`

For `x>1, sin ^{-1} frac{2 x}{1+x^{2}}`
`
=sin ^{-1} frac{frac{2}{x}}{1+frac{1}{x^{2}}}`
`
=2 tan ^{-1}(1 / x)=2 cot ^{-1} x`

Hence the given integral is
`
int_{0}^{sqrt{3}} frac{sin ^{-1}(frac{2 x}{1+x^{2}})}{(1+x^{2})} d x`
`
=int_{0}^{1} frac{2}{1+x^{2}}(tan ^{-1} x) d x+int_{1}^{sqrt{3}} frac{2 cot ^{-1} x}{1+x^{2}} d x`
`
=[(tan ^{-1} x)^{2}]_{0}^{1}-[(cot ^{-} x)^{2}]_{1}^{sqrt{3}}=frac{pi^{2}}{16}-(frac{pi^{2}}{36}-frac{pi^{2}}{16})`
`
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