Mathematics

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


SOLUTION
Let $$I=$$$$\displaystyle \int \sqrt{\dfrac{x - 5}{x - 9}}dx$$.
Let $$x-9=z^2$$......(1) then $$dx=2zdz$$
Using these in the above expression we get,
$$I=$$$$2\displaystyle \int \sqrt{z^2+4}dz$$
or, $$I=$$$$2\displaystyle \int \sqrt{z^2+2^2}dz$$
or, $$I=2\left(\dfrac{z\sqrt{z^2+4}}{2}+\dfrac{2^2}{2}\log|z+\sqrt{z^2+4}|\right)+c$$ [ Where $$c$$ is integrating constant]
or, $$I=\sqrt{(x-9)(x-5)}+4\log|\sqrt{x-9}+\sqrt{x-5}|+c$$. [ Using (1)]
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Subjective Medium Published on 17th 09, 2020
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