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