Question

Integrate the function: \(\frac {1}{\sqrt {(x-a)(x-b)}}\)

Answer 1

(x-a)(x-b) can be written as x² - (a+b)x + ab.
Therefore,
x²- (a+b)x + ab

= x²- (a+b)x + \(\frac {(a+b)^2}{4}\) - \(\frac {(a+b)^2}{4}\) + ab

= [x-(\(\frac {a+b}{2}\))]² - \(\frac {(a-b)^2}{4}\)

⇒ \(∫\)\(\frac {1}{\sqrt {(x-a)(x-b)}}\ dx\) = \(∫\frac {1}{\sqrt {{x-(\frac {a+b}{2})}^2-(\frac {a-b}{2})^2}} dx\)

Let x - (\(\frac {a+b}{2}\)) = t

∴ dx = dt

⇒ \(∫\frac {1}{\sqrt {{x-(\frac {a+b}{2})}^2-(\frac {a-b}{2})^2}} dx\) = \(∫\frac {1}{\sqrt {t^2-(\frac {a-b}{2})^2}}dt\)

= \(log \ |t+\sqrt {t^2-(\frac {a-b}{2})^2|}+C\)

= \(log \ |{x-(\frac {a+b}{2})}+\sqrt {(x-a)(x-b)}|+C\)