Evaluate the following integral: `int_0^(1//2)1/((1+x^2)sqrt(1-x^2))dx`

Explanation

Take `I=int frac{d x}{(1+x^{2}) sqrt{1-x^{2}}}`

Eliminate `x=frac{1}{t} Rightarrow d x=-frac{1}{t^{2}} d t`
`

I=int frac{-t d t}{(t^{2}+1) sqrt{t^{2}-1}}`


Eliminate `t^{2}-1=u^{2} Rightarrow 2 t d t=2 u d u`
`

I=-int frac{u d u}{(u^{2}+1) u}=-int frac{1}{u^{2}+(sqrt{2})^{2}} d u`
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