Question

Evaluate: $\int \sqrt{\frac{x-5}{x-9}}dx$.

Answer 1

Let $I=$$\int \sqrt{\frac{x-5}{x-9}}dx$.

Let $x-9=z^2$......(1) then $dx=2zdz$

Using these in the above expression we get,

$I=$$2\int \sqrt{z^2+4}dz$

or, $I=$$2\int \sqrt{z^2+2^2}dz$

or, $I=2(\frac{z\sqrt{z^2+4}}{2}+\frac{2^2}{2}\log |z+\sqrt{z^2+4}|)+c$ [ Where $c$ is integrating constant]

or, $I=\sqrt{(x-9)(x-5)}+4\log |\sqrt{x-9}+\sqrt{x-5}|+c$. [ Using (1)]