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

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

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

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