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

Explanation

Given

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

Putting `x=frac{1}{t}`

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

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

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

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

Putting `1+t^{2}=u^{2}`

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