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Integrate the function
$\int \frac{1}{x \sqrt{a x-x^2}} d x=I(s a y)$
$\int \frac{1}{x \sqrt{a x-x^2}} d x=I(s a y)$
Solution:
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Verified Answer
Put $x=\frac{a}{t}$ so that $d x=\frac{-a}{t^2} d t$
$\therefore \quad I=-\frac{1}{a} \int \frac{1}{\sqrt{t-1}} d t=\frac{-2}{a} \sqrt{\frac{a-x}{x}}+c$
$\therefore \quad I=-\frac{1}{a} \int \frac{1}{\sqrt{t-1}} d t=\frac{-2}{a} \sqrt{\frac{a-x}{x}}+c$
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