Question

Integrate the function: \(\frac {x-1}{\sqrt {x^2-1}}\)

Answer 1

\(∫\frac {x-1}{\sqrt {x^2-1}}\  dx\) = \(∫\frac {x}{\sqrt {x^2-1}}\  dx\) - \(∫\frac {1}{\sqrt {x^2-1}}\  dx\)            ……....(1)

\(For ∫\frac {x}{\sqrt {x^2-1}} dx, \ let x^2-1 = t ⇒ 2x dx = dt\)

\(∴ ∫\frac {x}{\sqrt {x^2-1}} dx = \frac 12 ∫\frac {dt}{\sqrt t}\)

\(=\frac 12 ∫t^{-\frac 12} dt\)

\(=\frac 12[2t^{\frac 12}]\)

\(=\sqrt t\)

\(=\sqrt {x^2-1}\)

\(From\  (1), we\  obtain\)

\(∫\frac {x-1}{\sqrt {x^2-1}}\  dx\) = \(∫\frac {x}{\sqrt {x^2-1}}\  dx\) - \(∫\frac {1}{\sqrt {x^2-1}}\  dx\)         \([∫\frac {1}{\sqrt {x^2-a^2}} \ dt = log\ |x+\sqrt {x^2-a^2|}]\)

\(= \sqrt {x^2-1}-log|x+\sqrt {x^2-1}|+C\)