Evaluate: `int1/((1+sqrt(x))sqrt(x-x^2))\ dx`

Explanation

`I=int frac{d x}{(1+sqrt{x}) cdot sqrt{x} sqrt{1-x}}`


Put `1+sqrt{x}=t Rightarrow frac{1}{2 sqrt{x}} dx=dt`


`Rightarrow I=int frac{2 d t}{t sqrt{2 t-t^{2}}}`


Again put `t=frac{1}{z} Rightarrow dt=frac{-1}{z^{2}} dz Rightarrow I=2 int frac{-frac{1}{z^{2}} dz}{frac{1}{z} sqrt{frac{2}{z}-frac{1}{z^{2}}}}=2 int frac{-dz}{sqrt{2 z-1}}`

`=-2 sqrt{2 z-1}+c `

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