Question

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

Answer 1

`" Let " I=int(1)/((x+1)sqrt(x^(2)-1))dx`
`" Putting " x+1=(1)/(t) " and " dx=-(1)/(t^(2))dt, " we get "`
`I=int(1)/((1)/(t)sqrt(((1)/(t)-1)^(2)-1))(-(1)/(t^(2)))dt`
`=-int(dt)/(sqrt(1-2t))=-int(1-2t)^(-1//2)dt`
`= -((1-2t)^(1//2))/((-2)((1)/(2)))+C=sqrt(1-2t)+C`
`=sqrt(1-(2)/(x+1))+C=sqrt((x-1)/(x+1))+C`