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
⇒ \(∫\frac {1}{\sqrt {(x-\frac 32)^2-(\frac 12)^2}} dx\) = \(∫\frac {1}{\sqrt {t^2-(\frac 12)^2}} dt\)
= \(log\ |t+\sqrt {t^2-(\frac 12)^2}|+C\)
=\(log\ |(x-\frac 32)+\sqrt {x^2-3x+2}|+C\)