Question

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

Answer 1

(x-1)(x-2) can be written as x2-3x+2.
Therefore,
x2 - 3x + 2
= x2 - 3x +$\frac 94$ -$\frac 94$ + 2
=(x - $\frac 32$)2 - $\frac 14$
=(x - $\frac 32$)2 - ($\frac 12$)2
∴ ∫$$\frac {1}{\sqrt {(x-1)(x-2)}}\ dx= $∫\frac {1}{\sqrt {(x-\frac 32)^2-(\frac 12)^2}} dx$
Let x-$\frac 32$ = t
∴ dx = dt
⇒ $∫\frac {1}{\sqrt {(x-\frac 32)^2-(\frac 12)^2}} dx$ = $∫\frac {1}{\sqrt {t^2-(\frac 12)^2}} dt$
= $log\ |t+\sqrt {t^2-(\frac 12)^2}|+C$
=$log\ |(x-\frac 32)+\sqrt {x^2-3x+2}|+C$