Question

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

Answer 1

Correct Answer - `sqrt((x-1)/(x+1))+C`
`" 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`