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Answer
$$ \displaystyle\int \sqrt{{\left(\dfrac{3}{2}\right)}^{2}-{\left(x-\dfrac{5}{2}\right)}^{2}}\, \mathrm d x = {{9\,\arcsin \left({{2\,x-5}\over{3}}\right)+4\,x\,\sqrt{-x^2+5\,x- 4}-10\,\sqrt{-x^2+5\,x-4}}\over{8}} $$
Explanation
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